FERROMAGNETIC RESONANCE IN NICKEL AT LOW TEMPERATURES
by
Jeffrey MacLeod Rudd
B.Sc.(Honours), Simon Fraser University, 1981
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
in the Department
of
Physics
0 Jeffrey MacLeod Rudd November 1985
SIMON FRASER UNIVERSITY
1985
All rights reserved. This work may not be reproduced in whole or in part, by photocopy
or other means, without permission of the author,
-ii-
APPROVAL
Name : Jeffrey MacLeod Rudd
Degree: Master of Science
Title of Thesis: Ferromagnetic Resonance in Nickel at Low
Temperatures
Examining committee:
Chairman: R. H. Enns
- J. F. Cochran
Senior Supervisor
.. -- I /
B. Heinrich
External Examiner
Department of Physics
Colorado State University
Date Approved: November 29, 1985
PART l AL COPYR l GHT L l CENSE
I
I hereby g r a n t t o Simon Fraser U n i v e r s i t y t h e r i g h t t o lend
my t h e s i s , p r o j e c t o r extended essay ( t h e t i t l e o f which i s shown below) '
t o users o f t h e Simon Fraser U n i v e r s i t y L i b ra r y , and t o make p a r t i a l o r I I
s i n g l e cop ies o n l y f o r such users o r i n response t o a reques t f rom t h e
l i b r a r y o f any o t h e r u n i v e r s i t y , o r o t h e r educa t iona l i n s t i t u t i o n , on
i t s own beha l f o r f o r one o f i t s ysers . I f u r t h e r agree t h a t permiss ion
f o r m u l t i p l e copy ing o f t h i s work f o r s c h o l a r l y purposes may be g ran ted
by me o r t h e Dean o f Graduate S tud ies . I t i s understood t h a t copy ing
o r p u b l i c a t i o n o f t h i s work f o r f i n a n c i a l g a i n s h a l l n o t be a l lowed
w i t h o u t my w r i t t e n permiss ion.
T i t l e o f Thes is /Pro ject /Extended Essay
FERROMAGNETIC RESONANCE I N NICKEL
AT LOW TEMPERATURES
Author :
( s i g n a t u r e )
JEFFREY MacLEOD RUDD
( name 1
~erromagnetic resonance has been measured at 24 GHz in ( 1 10 )
nickel disks at 4 K and from 60 K to room temperature. Samples
had a nominal purity of 99.99% and a residual resistivity ratio
of 38. The applied field was in the plane of the sample and
measurements were made with the field along each of the three
principal axes ( 1 0 0 ) , ( 1 1 1 ) and ( 1 1 0 ) . The room temperature
linewidth was found to be isotropic within experimental
uncertainty, and the linewidth, AH, was found to be 360 Oe. The
experimental results indicated that the linewidth is anisotropic
at low temperatures. We found AHll0 > A H l l 1 ana AHloo for
temperatures below 200 K. At 4 K we found AHloo = 1620+50 Oe,
AHl1, = 1815+50 Oe and AHl1, = 2050+50 Oe. Kambersky has
suggested that the large increase in magnetic damping in Nickel
on cooling to 4 K is due to the presence of degenerate states at
the Fermi surface near the X points of the Brillouin zone. The
contribution of these states to the damping has been calculated
using a simple model of electrons and spin waves coupled via the
spin-orbit interaction. The results exhibited qualitatively the
temperature dependence of the damping but the calculated damping
parameters were approximately 1/200 those required by exper-
iment. The predicted anisotropy of the linewidth did not agree
with experiment. The calculation indicated that AHloo should be
greater than OH,,, by approximately 4% at 4 K. It is suggested
that the large sheets of the Fermi surface play an important
role in the magnetic damping of Nickel at low temperatures.
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ACKNOWLEDGEMENTS
I would like to thank John Cochran for his assistance and
patience with this work. I would also like to thank the members
of our research group Bret Heinrich, Ken Myrtle and Ken Urquhart
for their many contributions. Thanks also to Qiao Rongwen and
John Dutcher, to the members of my Committee Carl Patton and
Daryl Crozier, and to fellow grad students and otherwise Mike
Denhoff, Don.Hunter, Andrew Kurn, Bill McMullan and John
Simmons.
Finally, I wish to acknowledge the financial support of the
Natural Sciences and Engineering Research Council and of Simon
Fraser University.
TABLE OF CONTENTS
Abstract .................................................... iii Acknowledgements ............................................ iv
List of Tables .............................................. vii ............................................ List of Figures viii
1. INTRODUCTION ........................................... 1 ...................................... 1.1 Introduction 1
1.2 Historical Overview .............................. 17 THEORY
2.1 Introduction ...................................... 29 2.2 Calculation of the Absorption .................... 34 2.3 Arbitrary Orientation of the Magnetization ....... 70 2.4 The Anomalous Skin Effect and FMR ................ 90 EXPERIMENTAL DETAILS
3.1 Introduction .................................... 104 ......................................... 3.2 Samples 106
3.3 Experimental Observation of FMR ................. 108 ................ 3.4 The Cavity and the Sample Holder 119
3.5 The 24 GHz Microwave System ..................... 123 3.6 Measurements at other Frequencies ............... 136
4 . EXPERIMENTAL RESULTS AND DISCUSSION .................. 13'7 4.1 Introduction .................................... 137 4.2 Room Temperature Results ........................ 146
4.3 Results at 77 K .............................. ee157
4.4 Results at 4.2 K ................................ 167
-vi-
4.5 Results at Intermediate Temperatures ............ 182 ....................................... . 4 . 6 Discussion 187
5 . CALCULATION OF THE DAMPING PARAMETER ................. 202 5.1 Introduction .................................. *.202 5.2 The Model ..................................... 0.208 5.3 Calculation of the Damping ...................... 225 5.4 Results .......................................... 239
BIBLIOGRAPHY ....................*.......,.................. -247
LIST OF TABLES
2-1 The effective magnetocrystalline anisotropy fields for the .................... three crystal axes in the ( 1 1 0 ) plane 45
2-2 The effects of damping and exchange on FMR in Nickel at 24 GHz. room temperature .................................. 69
2-3 Representative numbers to indicate the importance of non-local effects in Nickel .............................. 95
.......................... 4-1 Material parameters for Nickel 142 .
4-2 Experimental results at room temperature ................ 147 4-3 Frequency dependence of FMR at room temperature ......... 149
4-4 Experimental results at 77 K ............................ 158 4-5 Experimental results at 4.2 K ........................... 168
...................... 5-1 Parameters for the X5 hole pockets 213
5-2 The constants entering the interaction ~amiltonian ...... 224
5-3 Expressions for the damping parameters G I and G2 ........ 241
5-4 Calculated values for the damping parameters G . and G2 .. 242
LIST OF FIGURES
2.1 The geometry for the calculation of the absorption ....... 33
2.2 The crystal axes in the [ 1701 plane ...................... 42
2.3 The angles used in the calculation of the magnetocrystalline anisotropy fields ........................................ 42
2.4 The geometry for the boundary value problem .............. 59
2.5 Calculated absorption and absorption derivative for Nickel at 24 GHz. room temperature .............................. 66
2.6 The angles used in calculating the absorption for arbitrary orientation of the applied' field ........................ - 7 3
2.7 Calculated variation of the direction of the magnetization with the field. in-plane ................................ 76
2.8 Calculated variation of the direction of the magnetization with the field. out-of-plane .......................... - 7 7
2.9 Calculated absorption in Nickel at 24 GHz. 4.2 K. with and without dragging ....................................... 85
............. 2.10 Plots to illustrate the effects of dragging 88
3.1 The 24 GHz microwave cavity and sample holder ........... 120
................. 3.2 Field configurations for the TEl12 mode 120
3.3 Schematic drawing of the 24 GHz microwave system ........ 124
3.4 Part of the 24 GHz microwave system ...................... 27
3.5 The bolometer ........................................... 133
4.1 Frequency dependence of the FMR linewidth at room temperature ............................................ 151
4.2 Experimental and calculated absorption derivatives. room temperature ............................................. 153
4.3 The angular variation of the resonance field. room ............................................. temperature 155
............... 4.4 Experimental absorption derivatives. 77 K 160
4.5 Experimental and calculated absorption derivatives. 77 K 161
4.6 The angular variation of the resonance field. 7 7 K ...... 163
4.7 The.angular variation of the FMR linewidth. 7 7 K ........ 166
4.8 Experimental absorption curves. 4.2 K ................... 171
4.9 (a)Comparison of the absorption for the two samples. 4.2 K (b)~omparison of the absorption as measured using the bolometer and with the microwave diode .................. 172
4.10 The absorption and the absorption derivative. 4.2 K .... 174
4.11 Comparison of calculation and experiment. 4.2 K. wavenumber dependent damping ....................................... 176
4.12 Comparison of calculation and experiment. 4.2 K. wavenumber independent damping ..................................... 177
4.13 The angular variation of the resonance field. 4.2 K .... 180
4.14 Experimental absorption curve. 4.2 K. 9.495 GHz ........ 181
........ 4.15 The variation of the linewidth with temperature 183
4.16 The variation .of the resonance field with temperature . . 185
4.17 Calculated variation o f the linewidth with temperature; wavenumber dependent damping ........................... 194
4.18 Calculated variation of the damping shift with temperature. wavenumber dependent damping ............................ 197
4.19 Calculated variation of the damping shift with temperature ......................................................... 198
............................ 5.1 The band structure of Nickel 210
5.2 The X5 hole pockets .................................... 214
5.3 The geometry for calculation of the damping parameter ... 218
5.4 The effective magnetocrystalline anisotropy fields due to the X5 hole pockets ..................................... 240
5.5 The damping parameter. ( 1 0 0 ) direction .................. 243
5.6 The damping parameter. ( 1 1 1 ) and ( 1 1 0 ) directions ....... 244
1. I NTRODUCT I ON
1.1 Introduction
In the work reported in this thesis we were interested in
the magnetic damping processes operating in Nickel at low
temperatures. These magnetic damping processes are largely
responsible for the Ferromagnetic Resonance (FMR) linewidth.
The temperature dependence of the FMR linewidth in pure Nickel
has been a subject of interest since 1966 when Bhagat and
Hirst[l] found that the linewidth increased with decreasing
temperature and reached a saturation value at approximately
20 K. ~ambersky[2] in 1970 suggested a damping mechanism for
Nickel in which the damping increased with an increase in the
electron lifetime. This work was extended to low temperatures
by Korenman and Prange[3,4]. They found that the magnetic
damping should increase with an increasing electron lifetime and
that the FMR linewidth should saturate at sufficiently long
electron lifetimes. This was the behaviour observed by Bhagat
and Hirst.
A qualitative argument, presented below, suggests that the
Kambersky-Korenman-Prange damping mechanism may lead to an FMR
linewidth which differs depending on whether the applied
magnetic field is parallel to a (loo), a ( 1 1 9 ) or a ( 1 1 0 )
crystal axis. The difference between the linewidths for the
different crystal axes should be greatest at low temperatures
where the linewidth had reached its saturation value.
We have attacked this possibility of an anisotropic FMR
linewidth at low temperatures in two ways. We have performed
the experiment and find that the linewidth is indeed
anisotropic. We have carried out a calculation of the damping
to be expected from certain electron states on the Fermi surface
of Nickel which were thought to be largely responsible for the
increase in damping at low temperatures. The results of this
calculation indicate that these states do not account for the
magnitude or the anisotropy of the linewidth observed in
experiment.
In this introductory chapter we give a very brief
description of FMR in metals to provide a background to the work
presented in this thesis. This description concludes with the
presentation of our results. The final section of this chapter
is a brief historical overview of experimental work relevant to
our work. TO extract information from an experimental FMR curve
it is necessary to compare the data with the results of a
calculation. The theory which is used for these comparisons is
the subject of Chapter 2. The experimental details are
discussed in Chapter 3. In Chapter 4 the experimental results
are presented and compared with the calculations of Chapter 2.
The calculation of the damping parameter is carried out in
Chapter 5.
Introduction to FMR
For a description of FMR we start with the simplest
possible situation: a free electron at rest in a uniform
magnetic field H,. In equilibrium the electron spin magnetic
moment, z, will lie parallel with the magnetic field. The spin
angular momentum will point opposite to the magnetic field since
the magnetic moment points in the opposite direction to the
angular momentum. If we start with the spin parallel to the
field, rotate it away from the field direction and let it go,
what happens? There is a torque on the electron spin of - - 7 = p x SO SO the spin will precess around the field direction
at a circular frequency o = yHo, where y is the gyromagnetic
ratio, y = gjel/2mc, the ratio of the magnetic moment to the
angular momentum. The precession is in a clockwise sense when
looking along the field direction, and the angle of precession
is a constant. For an electron y = 1.76x107 0e"sec" so that
in a magnetic field of 2 kOe, say, the precession frequency is
5.6 GHz, which is in the microwave range. If the electron
interacts with other objects the precessional motion may be
damped as energy is transferred to the surroundings. The angle
of precession will decrease with time so that the direction af
the spin will spiral in to the field direction. The
precessional frequency depends on the magnitude of the applied
field. If energy is supplied by a magnetic field which varies
in time with a fixed frequency o, and the magnitude of the
applied' field is varied the precessing spin will absorb energy
when the resonance condition, w = yH,, is satisfied.
We are interested in ferromagnetic materials - materials where Avogadro's number of electron spins Rave been welded into
one giant magnetic moment by the quantum mechanical exchange
interaction. What happens in this situation when the material
is placed in an applied field? First of all because of
electron spin-electron spin and electron spin-lattice
interactions the field the electron spins 'see' is not just the
applied field H,. The dipole-dipole interaction between the
spins leads to a demagnetizing field. Coupling between the
orbital motion of the electrons and their spins leads to
magnetocrystalline anisotropy torques, to magnetoelastic torques
and to dissipative torques. If the direction of the
magnetization changes rapidly in space the exchange interaction
can produce a torque on the magnetization. These effects must
be considered when analysing the behaviour of the ferromagnet in
a magnetic field.
The behaviour of the magnetization is described by the
Landau-Lifshitz equation[5]:
where @ is the magnetic moment per unit volume of the material,
and 'eff is the effective field acting on the magnetization
which includes applied fields, demagnetizing fields, and
effective fields describing the torques exerted by
magnetocrystalline anisotropy and exchange. The damping torques
are included as TD. This equation and the terms entering it are
discussed in Chapter 2.
When displaced from equilibrium the magnetization precesses
as in the free electron case. The natural frequency is no
longer yHo but depends on the sample shape, the direction of the
applied field with respect to the sample, magnetocrystalline
anisotropy, exchange and damping. For example, if a disk shaped
sample is used, with the applied field parallel to the plane of
the disk (the configuration used in our experiments) resonance
occurs when :
in the absence of magnetocrystalline anisotropy, exchange and
damping. A Ferromagnetic Resonance experiment consists of
irradiating a ferromagnetic sample in an applied field with
microwaves of a fixed frequency. The magnitude of the applied
field is varied and the energy absorbed by the sample as a
function of the field is measured. When the applied field
satisfies the resonance condition the energy absorbed becomes a
maximum.
The quantities which combine to determine the shape and
position of the FMR absorption line include the saturation
magnetization 4 M S , the g-factor, the magnetocrystalline
anisotropy constants MI, K2..., the exchange constant A , and the
damping torques, usually described by a phenomenological damping
constant G (Gilbert damping) or X (Landau-~ifshitz damping). In
practice these quantities are so entangled in determining the
value of the applied field at which resonance occurs, Him,, the
linewidth, AH, and the lineshape that it is often difficult to
extract values for the material parameters from an experimental
curve. However measurement of the FMR linewidth provides
information about the magnetic damping processes which cannot be
so easily obtained usingvany other technique.
In discussing FMR it is often useful to think in terms of
spin waves, which are collective modes of the electron spin
system. In the ferromagnetic ground state all the electron
spins are parallel. Deviations from this lowest energy
configuration may be described as the excitation of spin waves.
In particular the precessional motion of the magnetization may
be described in terms of spin waves, so that in this picture FMR
consists of the excitation of spin waves by the microwave field.
Spin waves are characterized by an approximately quadratic
dependence of frequency, w , on wavenumber, k. For spin waves
propagating at right angles to the magnetization (as in our
experiments) the dispersion relation is[6]: ,
where A is the exchange constant. Spin waves may be thought of
as quasi-particles which interact with other quasi-particles in
the system, for example with the conduction electrons in a
metal, or with phonons. Magnetic damping occurs when a spin
wave is annihilated in a collision with an electron or a phonon.
In an insulating medium the wavenumber of the microwave
field is k = 2a/Xo, where Xo is the free space wavelength. For
microwaves k is of the order of a few cm'l. FMR occurs when the
frequency and wavenumber (energy and momentum) of a spin wave
matches the frequency and wavenumber of the microwave field.
For k of the order of a few cm-' 2Ak2/MS < Ho at FMR and
resonance occurs when:
The FMR Linewidth in Metals
In pure metals the FMR linewidth is largely due to two
contributions: the exchange-conductivity mechanism, and what is
described as the intrinsic damping. The relative importance of
these two contributions differs from metal to metal. In Nickel
the intrinsic damping is the major factor. For example, at room
temperature and a microwave frequency of 24 GHz, approximately
300 Oe of the 320 Oe linewidth arises from the intrinsic damping
with approximately 20 Oe being due to exchange-conductivity. In
Iron however the exchange-conductivity line broadening is larger
than the width due to the intrinsic damping. At room
temperature and 24 GHz approximately 95 Oe of the 110 Oe
linewidth comes from exchange-conductivity broadening with
approximately 15 Oe being due to the intrinsic damping.
The Exchange-Conductivity Broadening
Penetration of microwaves into a metal is limited by the
skin effect. For a metal in which the electrical conduction may
be described by Ohm's law, j = ooe, where j is the current
density, e is the electric field and uo is the dc conductivity, .
the spatial variation of the microwave fields with distance, z,
into the metal is:
where 6 is the skin depth. The skin depth is, in CGS units:
where e is the speed of light and p is the permeability. For
typical metals at microwave frequencies the skin depth is of the
order of half a micron. The skin depth depends on the magnitude
of the applied field since the permeability changes as the
applied field is swept reaching a maximum at FMR (see
Chapter 2). The spatial variation of the microwave fields leads
to a spatial variation of the magnetization. The resulting
non-uniformity produces an exchange torque which acts to restore
the parallelism of the electron spins.
The Fourier spectrum of the field (1.5) consists of a
distribution of wavevectors, q, the real part of which exhibits
a maximum at q = 1/6, and has a width of roughly I / & . Typical
values of q at FMR are 10'-105 cm-', much larger than for an
insulator. Spin waves will be excited over a range of values of
the applied field, leading to a broadening of the resonance and
to a shift from (1.4) in-the field at which resonance occurs.
These effects increase with a decreasing skin depth, that is,
with an increasing conductivity. The combined effects of
exchange and conductivity on the resonance lineshape have been
discussed by Ament and Rado[7].
Since the conductivity of a pure metal increases with
decreasing temperature this exchange-conductivity broadening may
become large at low temperatures. At low temperatures the
conduction electron mean free path, I , may become comparable
with the skin depth (the 'anomalous skin effect regime'). If
this occurs Ohm's law is no longer valid and the conductivity
becomes 'non-local' or 'wavenumber dependent' (see Chapter 2 ) .
The skin depth is no longer ( 1 . 6 ) but instead saturates at a
value = (621)1'3[8]. Since the permeability is large at FMR,
the skin depth is small, and the wavenumber dependence of the
conductivity may become important at much higher temperatures
than for non-magnetic metals. The result of the wavenumber
dependence of the conductivity is that the exchange-conductivity
broadening is reduced from that which would be expected based on
the dc conductivity. The treatment of FMR using a wavenumber
dependent conductivity has been discussed by Hirst and
~range[91.
The Intrinsic Damping
The intrinsic damping has its origin in the coupling
between the electron spin system and the crystal lattice through
the spin-orbit interaction. In the presence of spin-orbit
coupling electron states are no longer spin eigenstates but
contain a mixture of up and down spin character. Scattering of
an electron, by phonons or impurities, may change the total spin
of the system resulting in magnetic damping. The major
contribution to the damping due to this mechanism comes from
'spin-flip' scattering in which an electron is scattered from a
state of predominantly one spin character to a state of
predominantly the opposite spin character, in a different energy
band. This inter-band scattering damping mechanism has been
considered by Elliott[lO] and Kambersky[Z]. The damping due to
this mechanism varies inversely with the electron relaxation
time, or, equivalently, varies as the electrical resistivity.
As a result of spin-orbit coupling the energy of an
electron state may be shifted from the energy in the absence of
spin-orbit coupling. In ferromagnetic materials the shift may
depend on the direction of the magnetization with respect to the
crystal axes. The effects of these energy shifts may be large,
for example, the shifts are the origin of magnetocrystalline
anisotropy, see for example, Kondorskii[ll]. In an FMR
experiment the precession of the magnetization causes the
energies of the electron states to vary periodically with time.
This variation is a source of magnetic damping. The simplest
way to picture this damping mechanism is to consider the
electron spin system and the electron system as distinct and to
consider the coupling between them introduced by the spin orbit
interaction. An electron on the Fermi surface and a spin wave
may collide because of this coupling with the annihilation of
the spin wave and scattering of the electron. Energy and
momentum must be conserved in such a collision. Energy
conservation restricts the scattering to states in the same
energy band ('intra-band scattering') because the spin wave
energy is much smaller than the electron kinetic energy. The
annihilation of a spin wave produces magnetic damping. This
damping mechanism was suggested by Kambersky[2] who showed that,
near room temperature, the damping should increase with the
electron relaxation time, that is it should increase as the
electrical conductivity. This work was extended to low
temperatures by Korenman and ~range[3,4]. They found that the
damping due to this mechanism became wavenumber dependent and
varied as the wavenumber dependent conductivity. For a
spherical Fermi surface they found that the Gilbert damping
parameter varied as:
where q is the wavenumber of the electromagnetic field in the
metal, and 1 is the electron mean free path. The mean free path
is related to the relaxation time by the Fermi velocity,
1 = v r . The linewidth due to this damping mechanism increases F with increasing relaxation time (decreasing temperature)
eventually reaching a constant value., independent of the
relaxation time, in the limit ql >> '1 (the extreme anomalous
limit).
Intra-band Scattering in Nickel
The increase with decreasing temperature and eventual
saturation of the binewidth is the behaviour that was observed
in pure Nickel at bow temperatures and 22 GHz by Bhagat and
Hirst[l] and Bhagat and ~ubitz[l2,13]. They found that the FMR
linewidth increased with decreasing temperature starting at
approximately 150 K and reached a plateau at approximately 20 K
of roughly five times the room temperature value. The value of
the linewidth at this plateau was found to be independent of the
residual resistivity ratio ( p 2 , , / p 4 ) of the samples if this
ratio was greater than 30. The increase in linewidth was much
larger than could be accounted for by any increase in the
exchange conductivity broadening due to the increased
conductivity.
Lloyd and ~hagat[l4] performed experiments with
Nickel-Copper alloys to test the dependence of the increase in
the linewidth on the resistivity ratio. It was found that the
increase in the linewidth disapppeared in a 5.4% Cu-~i alloy
which had a resistivity ratio of 3. The magnetization and other
magnetic properties were not much affected by alloying so the
results were interpreted as evidence that the increase in the
linewidth and the increase in the electron relaxation time were
related.
Further evidence for the wavenumber dependence of the
damping came from the measurements of Heinrich, Meredith and
~ochran[l5] in which the transmission of microwaves through thin
samples of Nickel was measured as a function of the applied
field. At the field corresponding to Ferromagnetic
Antiresonance (FMAR) the permeability is small (in the absence
of damping it is zero). The skin depth at FMAR is much larger
than at FMR so that the typical wavenumbers are much smaller at
FMAR than at FMR. The damping parameters deduced from the FMAR
data were much larger than those quoted by Bhagat and ~ubitz[42]
.in the same temperature range. This is what would be expected
from a damping of the form of equation (1.7) since arctan(ql)/q
decreases with increasing q.
In'Nickel a large contribution to this intra-band damping
is conjectured to come from d-band minority spin states near the
X-points in the Brillouin zone[2] (the X-points are at the zone
edge in the (100) directions). The Fermi surface from these
bands is a set of three approximately ellipsoidal surfaces
centered at the X-points with the long axis of the ellipse along
the axis connecting X with the center of the Brillouin zone.
These surfaces are called the X, hole pockets. In the absence
of spin orbit coupling the X, level is doubly degenerate. The
degeneracy is lifted by the spin orbit interaction, the
splitting between the two levels depending on the direction of
the magnetization with respect to the crystal axes[l6]. As a
result the size and the shape of the pockets depends on the
direction of the magnetization. For example with the applied
field along ( 1 1 1 ) all three pockets are equivalent. With the
magnetization along (100) the pocket along the field direction
is smaller in all dimensions than the pockets that are at the
X-points perpendicular to the field direction.
In a collision between an electron and a spin wave energy
and momentum must be conserved. At low temperatures, long
electron relaxation times, conservation of momentum restricts
the electrons which may collide with a spin wave to those whose
velocity is approximately perpendicular to the direction of spin
wave propagation (see Chapter 5). This leads to the idea of an
'effective zonev of the Fermi surface consisting of those
electrons which may interact with spin waves. Since the size
and shape of the pockets depends on the direction of the
magnetization with respect to the crystal axes the electrons
comprising the 'effective zone' will differ depending on the
direction of the magnetization. The result may be a dependence
of the magnetic damping on the direction of the magnetization
which would be manifested as an anisotropy of the FMR linewidth.
The Work Reported in this Thesis
This possibility of an anisotropic linewidth at low
temperatures has been investigated in two ways. Ferromagnetic
resonance was measured in pure Nickel samples with the applied
field parallel to each of the three principal crystal axes
(loo), (111) and (110). The resistivity ratio of the samples
was 38 so that we were above the limit of approximately 30 for
which the linewidth was found to reach its limiting value[l].
Because the FMR line becomes very broad on cooling,
approximately 1600 Oe at 24 GHz at 4.2 K t the signal becomes
small and difficult to detect. We were unable to observe FMR
using a conventional field modulation technique at temperatures
below approximately 60 K. We used a bolometer to detect the
absorption at 4.2 K. Our results indicate that the linewidth is
anisotropic at low temperatures. For temperatures below
approximately 200 K the (110) linewidth is greater than the
(100) and ( 1 1 1 ) linewidths. The (100) and (111) linewidths were
the same for temperatures above 60 K, the lowest temperature at
which we could measure FMR using field modulation. This is the
same behaviour observed by Anders, Bastian and ~iller[l7] for
temperatures greater than 77 K. At 4.2 K, the only temperature
at which the bolometer could be used, we found that the (100)
linewidth was 1620250 Oe, the ( 1 1 1 ) linewidth was 1820+50 Oe and
the (110) linewidth was 2050+50 Oe.
A calculation of the damping parameter has been carried out
using a simple model of electrons and spin waves coupled by the
spin orbit interaction. We calculated the susceptibility using
the method of Green's functions. The imaginary part of this
susceptibility was related to the damping parameter. The
expression we obtained for the damping parameter reduced to that
obtained by Korenman and ~range[3,4] if we assumed a spherical
Fermi surface. The integrals over the Fermi surface which enter
the damping parameter were evaluated numerically using the
description of the X, hole pockets of Hodges, Stone and
Gold[16]. The results were not in agreement with experiment.
The values of the damping parameter were approximately 100 times
too small to account for the linewidth in Nickel at 4.2 K. The
calculated anisotropy of the damping parameters was also not in
agreement with experiment. For example, based on the
calculation we expected that the linewidth for the (100)
direction should be approximately 4% larger than the linewidth
for the (111) direction. From experiment we found that the
( 1 1 1 ) linewidth was approximately 12% larger than the (100)
linewidth. The implication of these results is that other
portions of the Fermi surface must play an important role in the
magnetic damping of Nickel at low temperatures.
1.2 Historical Overview
We present a brief historical overview of the experimental
work relevant to the work presented in this thesis. Most of the
work cited involves measurements of FMR on Nickel at room
temperature and below. The order of presentation is as follows:
first the most important FMR measurements are mentioned: then a
series of measurements relating to the anisotropy of the FMR
linewidth of Nickel which are conveniently discussed as a group,
although made concurrently with the first set of experiments.
As an example of the treatment of the anomalous skin effect in
FMR we discuss a measurement on Iron. Finally FMAR measurements
of the damping in Nickel are discussed.
FMR in Nickel
Measurements of the properties of ferromagnets at microwave
frequencies have been made since the turn of the century. A
review of work done before 1950 is given by ~ado[l8].
The first measurement of FMR was made by ~riffiths[l9] in
1946 on samples of Nickel, Iron and Cobalt. is samples were
thin films (0.025 mm) electrodeposited on a brass disk. The
Pinewidth for Nickel was approximately 2000 Oe at 9 GHz, This
is much larger than the linewidth for a good single crystal of
Nickel at this frequency (approximately 100 Oe, see below).
Low temperature measurements were made in 1956 by
Reich[20]. The measurements were made on (110) plane single
crystal disks at frequencies of 9 and 24 GHz, at 4.2 K, 77 K and
room temperature. The 'half-line-width' at 24 GHz was of the
order of 350 Oe and independent of temperature. From the
resonance field values a value of the g-factor, g = 2.21f0.01,
and values for the magnetocrystalline anisotropy constants K1
and K2 were obtained. The values of K1 were -5.4x105 and
-8.3x105 erg/cm3 at 77 and 4.2 K respectively. These are
approximately two-thirds- the accepted values of -8.45x10f and
-12.9x105 [211. The samples had a poor resistivity ratio,
~ 2 9 5 / ~ 4 = 7. This low purity explains the temperature
independent linewidth and may also account for the discrepancy
in the values of the anisotropy constants. Franse [22] has
shown that the anisotropy constants of Nickel may be sensitive
to impurities.
The first measurements in which the intrinsic linewidth of
Nickel was observed, as opposed to that due to the sample
quality or surface preparation techniques, were made by
Rodbe11[23,24] in 1964 on Nickel platelets and whiskers.
Platelets and whiskers are small, very perfect, single crystals
grown by a vapor reduction process, The virtue of these samples
is that the surfaces are smooth and require no polishing or
other preparation. Thus the measurements reflect as closely as
possible the ideal behaviour of the material. The samples are
small and very fragile and require great care in handling. This
point is discussed further below. Rodbell's measurements were
made at 9 and 35 GHz at temperatures from 130 K to 635 K (the
Curie Temperature of Nickel). A t room temperature the 9.2 GHz
linewidth was typically 130 Oe. The results were described by a
Landau-Lifshitz damping parameter X = 2.5x108 sec-I independent
of temperature and frequency. This parameter varied slightly
from sample to sample. A frequency and temperature independent
value for g of 2.2220.03 was found. No spin-pinning was
required to match the results. values for K1 and K2 were
deduced over the temperature range. The values of K1 were in
agreement with those of other authors. The values of K2 are in
agreement if the correction pointed out by ~ubert[25] is taken
into account. These measurements are probably the only
worthwhile FMR results reported on Nickel platelets.
The frequency dependence of the FMR linewidth at room
temperature was investigated in 1965 by Frait and ~acFaden[26].
Measurements were made on a number of materials including single
crystals of pure Nickel. These samples were disks spark cut
from a bulk single crystal, mechanically polished, annealed for
several hours, then electropolished. The linewidth at 25 GHz
was 520 Oe. This is much larger than the intrinsic linewidth ~f
330 Oe expected at this frequency for a good sample. *The
frequency dependence for Nickel (8 to 72 GHz) could not be
explained by any macroscopic theory. The large linewidths were
ascribed to inhomogeneities and stresses in the sample. The
lesson to be drawn from these results is that it is not
straightforward to reproduce the linewidth characteristic of the
metal in bulk samples. A value of g = 2.2120.04 was found.
Values of K1 and K2 in agreement with accepted values were
obtained.
Bhagat, Hirst and ~nderson[27] made measurements similar to
those of Frait and MacFaden in 1966. heir samples were (110)
plane disks and cylinders oriented with the cylinder axis
parallel to either the ( 1 1 1 ) or (100) axis. These samples were
spark cut from bulk single crystals and electropolished. The
linewidth at a frequency-of 21.7 GHz was 300 Oe. From the
frequency dependence of the linewidth(9 to 57 GHz) they found
the Landau-Lifshitz damping parameter to be 2.3x108sec-'. It
was found necessary to use a surface anisotropy Ks=0.25erg/cm3
with the anisotropy axis parallel to the sample surface to
reproduce the experimental results. The value of the g-factor
used in the analysis* was 2.22.
These measurements were extended to low temperatures by
Bhagat and ~irst[l] in 1966, in the first good low temperature
measurements on Nickel. Measurements were made over the
temperature range 4.2 K to 300 K at 9, 22 and 35 GHz. They
found that the linewidth increased on cooling and that the 4.2 M
linewidth was independent of the resistivity ratio in samples
with resistivity ratios of 30, 60 and 160. The observed
linewidths at 4.2 K were 640 Oe at 9 GHz, 1500 Oe at 22 GHz and
2200 Oe at 34.8 GHz. These linewidths were much larger than
could be explained by the exchange conductivity mechanism. The
results were interpreted as evidence for a temperature
dependence of the damping parameter.
In 1969 ~ranse[28] made FMR measurements on Nickel in an
attempt to measure the magnetocrystalline anisotropy constants.
He used (110) plane disks electrically or mechanically polished.
The linewidths were quite large, being 600 Oe at room
temperature and 1200 Oe at 77 K at a frequency of 23.3 GHz. The
values of K1 and K2 deduced from the FMR data at 77 K were in
agreement with the values obtained using a torque magnetometer.
Franse comments that the-accuracy of FMR measurements of the
magnetocrystalline anisotropy constants is much smaller than may
be obtained from torque experiments. He also states that it is
impossible to obtain values for the higher order constants, K3,
K4..., using FMR. This is a problem because these higher order
constants are important in Nickel at low temperatures.
The connection between the increase in the linewidth at low
temperatures and the increase in the conduction electron mean
free path was made in 1970 in the experiments of Lloyd and
Bhagatil41. The temperature dependence of the linewidth at
35 GHz was measured using cylindrical samples oriented with a
(100) or a ( 1 1 1 ) axis along the cylinder axis. The samples used
were pure Nickel and 0.17% and 5.4% Copper in Nickel alloys.
The linewidth of the 0.17% Cu alloy (resistivity ratio of 30) at
as approxim ately 90% that of pure Nickel, while the 5.4%
Cu alloy (resistivity ratio of 3) showed no increase in the
linewidth with decreasing temperature. A value of I K I I / M ~ of
2150 G for Nickel at 4.2 K is quoted. This is based on "the
shift in the resonance field as a function of temperature". The
currently accepted value of I K I I / M ~ is 2460 G (see the
discussion of Nickel parameters in Section 4.1). There is a
fair discrepancy between these two values. We will discuss our
results on the position of FMR in Chapter 4 and argue that this
discrepancy is due to the wavenumber dependence of the magnetic
damping.
In 1971 Anders, Bastian and Biller[l7] made measurements on
(110) Nickel disks. Thevsamples were cut from 99% pure Nickel,
electropolished and carefully anndaled. FMR was measured for
different directions of the applied field with respect to the
crystal axes in the sample plane. Measurements were made at
9.2, 19.6 and 26.2 GHz. at temperatures between 77 K and 630 K.
The linewidth at 26.2 GHz at room temperature was approximately
350 Oe (this value was taken from their Figure 1). They found
the Landau-Lifshitz damping parameter to be 2.3x108 sec-'
independent of temperature and orientation of the applied field
for temperatures from 273 to 630 K. Below 273 K they found that
the linewidth was anisotropic, with the (110) linewidth being
greater than the (100) and ( 1 1 1 ) linewidths. The difference in
the linewidths for the different crystal axes increased with
decreasing temperature. From their Figure it appears that the
difference between the (10Q) and ( 1 1 1 ) linewidths was not
significant. At 77 K, at 26.2 GHz the (110) linewidth was
820 Oe and the ( 1 1 1 ) linewidth was 640 Oe. A value for the
(100) linewidth at this frequency is not quoted.
In 1974 Bhagat and Lubitz[l2,13] reported the results of
further experiments on Nickel at low temperatures. The
experiments were aimed at defining the temperature dependence of
the linewidth between 77 K and 4.2 K where the linewidth attains
its saturation value. The samples were cylinders with a ( 1 1 1 )
axis along the cylinder axis. These were electropolished and
annealed. Values of the Landau-Lifshitz damping parameter
obtained by comparing the experimental data with the calculation
of Hirst and Prange[9] which includes a non-local conductivity
are quoted. A value of g = 2.22 and a surface anisotropy of
0.1 erg/cm2 were used in the analysis. It is important to note
that in none of the work reported by Bhagat and colleagues is
there any mention of an anisotropy in the linewidth at low
temperatures. We will come back to this point when we discuss
our results in Chapter 4.
Anisotropy of the Linewidth
We discuss now a series of measurements related to the
anisotropy of the FMR linewidth. In 4967 Vittoria, Barker and
Yelon[29] made measurements on Nickel platelets of the
dependence of the FMR Pinewidth on the direction of the applied
field with respect to the crystal axes. They made measurements
by rotating the applied field both in the sample plane and out
of the sample plane. Their results were interpreted as evidence
that the damping parameter was anisotropic.
In response to these measurements Anderson, Bhagat and
Cheng[30] in 1971 reported similar measurements made on single
crystal disks cut from a bulk single crystal. Measurements were
made on (100) and (110) normal disks at 22 GHz. The linewidth
was found to be isotropic for in-plane variations of the
direction of the applied field, within the experiemntal
uncertainty of +I0 Oe. The linewidth for out of plane
variations of the direction of the applied field was anisotropic
because of the misalignment obetween the magnetization and the
applied field caused by the demagnetizing field. APP the
results were consistent with an isotropic damping parameter.
Maksymowicz and ~eaver[31], also in 1971, made similar
measurements on Nickel platelets, both in and out of the sample
plane. The dependence of the linewidth on angle was described
as 'a sum of a constant term and a term resulting from the
Gilbert type equation of motion'. The damping parameter
required was isotropic.
In 1972 ~ailey and ~ittoria[32] reported further
measurements on platelets. The measurements were of the
in-plane variation of the linewidth at 9.4 GHz at temperatures
from 171 to 293 K, The results indicayed that the (100)
linewidth was larger than the (110) linewidth at all
temperatures and that the difference between the linewidths for
the two directions increased on cooling. The room temperature
linewidth was 165 Oe which was larger by approximately 30% than
the linewidths found by Rodbe11[24] at a similar frequency.
Experiments using Nickel platelets are very difficult
because of the extreme fragility of the samples. Almost any
handling at all will damage the platelet resulting in broadening
of the FMR line. This was demonstrated very clearly by Wu,
Quach and Yelon[33] who made measurements on Nickel-Cobalt
platelets. Nickel Cobalt platelets are thicker, and hence more
robust, than pure Nickel platelets. However even with careful
handling of the samples the results showed anomalies which could
be explained only as a consequence of damage to the samples due
to handling. It appears then that the anisotropy of the
linewidth observed by Vittoria et a1[29,32] resulted from the
use of damaged samples rather than as a result of an anisotropy
of the intrinsic damping.
Vittoria et a1[34,351 were also responsible for two papers
in which the angular variation of the linewidth was calculated.
In the second of these papers the claim is made that the
combination of exchange and magnetocrystalline anisotropy may
produce an anisotropy of the linewidth. This is true if the
linewidth is almost entirely due to exchange conductivity
broadening and if the conductivity is large. The authors
suggest that they performed calculations appropriate for Nickel,
however they use a Landau-Lifshitz damping parameter of
/
0.375x10hecc" which is much smaller than the currently
accepted value of 2.45~10' sec- '. We have performed
calculations in which both exchange and magnetocrystalline
anisotropy are included using realistic parameters for Nickel at
a temperature corresponding to 77 K with a local conductivity
and we find no difference between the calculated linewidths for
the (loo), ( 1 1 1 ) or the (110) directions.
The Anomalous Skin Effect
At low temperatures the wavenumber dependence of the
conductivity becomes important and must be considered in a
calculation carried out for comparison with experiment. A
calculation of the FMR absorption with a non-local conductivity
was carried out by ~ i r s t and ~range[9]. A computer program
based on their calculation was used in the analysis of the data
of Bhagat and Hirstil] and Bhagat and ~ubitz[12,13]. The
linewidth in Nickel is dominated by the intrinsic damping as
stated in Section 1.1. Measurements on Iron whiskers at low
temperatures made in 1967 by Bhagat, Anderson and Wu[36]
provided a test of the Hirst-Prange theory in a material in
which the linewidth is predominantly due to exchange
conductivity broadening. The experimental results for Iron were
in good agreement with the theory, which predicted that the
exchange conductivity broadening should increase much less
rapidly with decreasing temperature (increasing conductivity)
than would be expected if a theory using the dc conductivity
were used. In addition to broadening the FMR line, exchange
conductivity produces a shift in the value of the applied
magnetic field at which resonance occurs. Calculated values of
this shift as a function of resistivity ratio for Iron are
quoted in the paper. Unfortunately the shifts in the resonance
field due to magnetocrystalline anisotropy were not known well
enough to permit a comparison between the experimental and
calculated resonance positions.
FMAR Measurements in Nickel
The microwave permeability is large at FMR with the result
that the skin depth is small. As stated above typical
wavenumbers at FMR are l o 4 - 1 0 5 cm-'. At the value of the
applied field corresponding to Ferromagnetic Antiresonance(FMAR,
see Chapter 2) the permeability is small so that the skin depth
becomes large. Typically wavenumbers at FMAR are 1/20 those at
FMR. Measurement of the transmission of microwaves at FMAR
through thin samples provides a very sensitive measure of the
damping parameter, see for example Cochran and ~einrich[37].
Since this is a transmission technique it is sensitive to the
bulk of the sample as opposed to FMR which is sensitive to a
surface layer approximately one microwave skin depth thick.
Transmission measurements through polycrystalline Nickel
foils by Dewar, Heinrich and Cochran[38] yielded values of
G = 2.45f.1x108 sec-I for the damping parameter at room
temperature and a value of g = 2.187k.005 for the 9-factor.
These may be considered the definitive values for these
parameters.
Low temperature measurements were made on single crystal
samples by Heinrich, Meredith and Cochran[l51 and by Myrtle1391.
The damping parameter was found to increase with decreasjng
temperature below approximately 250 K. The increase was much
more rapid than the increase in the damping parameter found in
the FMR measurements of Bhagat and ~ubitz[12]. This was
interpreted as evidence for the wavenumber dependence of the
damping. The temperature dependence of the FMAR damping
parameter was described well by the expression:
where oo and p are the dc conductivity and resistivity
respectively, and a and b were constants chosen to match
experiment. The values of a and b which fitted the temperature
dependence were a = 1.07x108sec'' and b = 1.19x108sec''. The
first of these terms corresponds to the result of Korenman and
~range[3,4] with the substitution arctan(ql)/ql 1 since the
values of q are small at FMAR. The second term corresponds to
the result of ~lliott[lO] and ~ambersky[2] for spin-flip
scattering .
2. THEORY
2.1 Introduction
Two different problems are of theoretical interest in
connection with the work reported in this thesis. First, the
phenomenological theory which is used to deduce fundamental
magnetic parameters from experiment, and second, the microscopic
theory which can be used to calculate a value of the damping
parameter, G I from the band structure for comparison with the
value of G deduced from the data and the phenomenological theory
used to describe FMR absorption. The phenomenological theory is
described in this chapter; a discussion of the calculation of
the damping parameter is postponed until Chapter 5.
The quantity measured in a ferromagnetic resonance
experiment is either the absorbed power as a function of the
applied magnetic field, or the derivative of the absorbed power
with respect to the field. The resulting curve is described by
the resonance field, Hfmr, the value of the applied field at
which the absorbed power is a maximum, and by the linewidth, AH,
the field interval between the extrema of the derivative of the
absorption with respect to field.
The position and shape of an FbrIR absorption line depend
upon a number of factors. Material parameters include the
saturation magnetization, M,, the spectroscopic splitting
factor, 9, magnetocrystalline anisotropy constants, Kt, K2, the
exchange constant, A, the damping parameter, G I and the
electrical conductivity, oo. The microwave frequency and the
sample shape also play a role. These factors are so entangled
in determining the position and linewidth that to extract values
for material parameters from an experimental line it is.usually
necessary to compare the data with an FMR lineshape calculated
using a phenomenological theory. Frequency and temperature
dependences of the resonance field and linewidth often prove
useful in sorting out the various contributions. Care must be
taken when fitting the observed lineshapes as these can be
easily affected by experimental factors which do not affect
Hfmr or AH. For example the asymmetry of an experimental
derivative line, the ratio of the high field derivative peak to
the low field derivative peak, often differs from that expected
from a calculation while the linewidth and position are close to
those expected (see Bhagat, Hirst and Anderson[27]).
Three calculations are described in this chapter. These
correspond to the variety of situations with which we are faced
experimentally. In the first calculation, Section 2.2, the
standard FMR treatment is presented. The applied field is taken
to be parallel to the sample plane and also parallel to one of
the three principal crystal directions ( 1 0 0 ) ~ (110), (1 1 1 ) . A
local electrical conductivity is assumed, ie 3 = o, Z , where
is the current density, G the electric field and oo the dc
conductivity. This calculation is appropriate for temperature
regimes where the approximation of a local conductivity is
valid. For Nickel this approximation is valid for temperatures
above approximately 77 K. The effects of exchange are included.
This is the simplest geometry to treat as the magnetization is
parallel to the applied field at the fields at which resonance
occurs, and it corresponds to the geometry which is used.in an
experiment.
If the applied field is allowed an arbitrary orientation
with respect to the sample plane and the crystal axes,
magnetocrystalline anisotropy and demagnetizing effects combine
so the static magnetization is not, in general, parallel to the
applied field. The angle between the magnetization and the
field depends on the magnitude of the field. The magnetization
is said to 'drag' behind the field. In an experiment an attempt
is made to align the sample such that the external field is
applied exactly in the sample plane and exactly parallel to a
crystal axis. If these conditions are not met dragging will
occur. The observed lines will differ from those expected on
the basis of the calculation of Section 2.2. To obtain an idea
of the magnitude of the discrepancy introduced by misalignment a
calculation is carried out in Section 2.3 which allows for
arbitrary orientation of the applied field relative to the
sample plane, as well as relative to the crystal axes. A local
conductivity is assumed, Exchange is not included as the
calculation becomes quite complicated. This is not a major
shortcoming since for Nickel exchange torques are small compared
with the damping torque.
At low temperatures (below 77 K) the increase in conduction
electron mean free path results in the conductivity and the
damping becoming wavenumber (q) dependent. A different approach
for a calculation of the absorption is required. These effects
are considered in the third calculation, outlined in ,
Section 2.4. The geometry is the same as in the first
calculation the applied field being in the plane of the sample
and along a principal axis. Exchange is included.
Experimental Geometry
In the experiments reported in this thesis the sample was a
Nickel single crystal in the form of a thin disk cut with a
( 1 1 0 ) axis normal to its plane. The sample formed part of the
endwall of a cylindrical microwave cavity. The applied field
was oriented parallel to the sample plane (the 'parallel
configuration') and could be rotated in that plane. The sample
was attached to a demountable endplate so that experiments eoubd
be performed for the applied field parallel to the different
crystal directions but with the microwave magnetic field
maintained perpendicular to the static magnetic field.
For the calculations outlined in this chapter the geometry
of Figure 2.1 is assumed. The sample forms an infinite slab
lying in the x-y plane, with the sample normal along [ 1 i 0 1 : in
Figure 2.1 The geometry used for the calculation of the
absorption. The sample lies in the x-y plane. The geometry
shown is for a calculation with the applied field parallel to
- the [ 0 0 1 ] axis. The sample may be rotated about the z-axis so
that the [ I 1 1 1 or the [ 1 1 0 ] axes can be oriented parallel to the
x-axis.
consequence the [OOl], [110] and [ 1 1 1 ] directions lie in the
sample plane (see Figure 2.2). For the first and third
calculations (Sections 2.2 and 2.4) the applied field points
along the x-direction and the crystal is oriented with a
principal axis in the x-direction.
Linearly polarized microwaves, with the microwave magnetic
field along the y-direction, propagate in the z-direction and
are incident on the sample at z = 0. The slab thickness is much
larger than the microwave skin depth so the sample may be
treated as semi-infinite. The slab is taken to be in free
space. It can be shown (see, for example, Urquhart[40]) that
the results for the calculation with boundary conditions
corresponding to a sample in a cavity differ only by a scaling
factor from those of the free space calculation.
2.2 Calculation of the Absorption
This calculation breaks into two distinct pieces. First
the microwave permeability is found by solving the
Landau-Eifshitz equation, This permeability is then combined
with Maxwell's equations to solve the boundary value problem of
. reflection from the metal surface yielding the absorbed power as
a function of applied field.
The Landau-Eifshitz equation (often termed the equation of
motion) is simply the statement that the rate of change of
angular momentum is equal to the torque. The Landau-Lifshitz
equation may be written:
where is the magnetic moment per unit volume,
y = glel/2mc = (g/2)(1.7588x107 Oe-' sec-'1 is the gyromagnetic
ratio, (the ratio of magnetic moment to angular momentum for an
electron), the g-factor for ~ickel is g = 2.187+.005[38], and 7
represents the torques acting on tRe magnetization.
In equilibrium the magnetization is parallel to the applied
field. Only small deviations from equilibrium will be
considered and therefore the magnetization can be written:
where idS is the saturation magnetization (parallel to the
applied field) and Z(2.t) is the deviation of a from as. The magnitude of is taken to be much less than Ms. All quantities
are assumed to vary in the z-direction only, with a time and
space dependence exp( i ( kz-wt ) ) so Z(?, t = iii exp( i ( kz-ot ) ) . The torques acting on the magnetization are due to
( i ) static and microwave applied fields; (ii) demagnetizing
fields; (iii) magnetocrystalline anisotropy; (iv) exchange;
(v) magnetostriction; and (vi) damping. These are discussed
individually below.
The concept of an 'effective field' proves important in
what follows to describe the torques acting on the
magnetization. A brief discussion is given here. For more
detail the reader is referred to 'Micromagnetics' by
W.F. ~rown[$l].
Three components are required to define the vector a. These may be the components in a rectangular coordinate .system
(Mx,M ,MZ), or the components in a spherical-polar coordinate Y
system (M,8,4) where M is the magnitude of a, 8 and t$ are the
polar and azimuthal angles respectively. The magnitude of the
magnetization is fixed, M = MS, so only the angles are
independent. If the energy of the system, magnetization and
surroundings, is writtenFE(8,t$) the components of the torque on
the magnetization are -aE/a8 and -aE/at$. It is usually more
convenient to work with the components (M~,M ,MZ). The vector Y
B = -aE/aW may be regarded as an 'effective field' and the
torque found from 7 = a x B. The energy may be written
E(Mx,M M ) or, taking into account the constraint on the length Y'
of a. E(Mx,M ~M:-(M:+M~)). The effective fields obtained from Y' Y
these two ways of writing the energy will be different, however
the torques will be the same.
A simple example might be useful. Consider an applied
magnetic field along the z-axis. The energy is E = -A*Bo. Let
E l = -MZHo and E2 = -(JM;-(M:+M~))H~. These expressions are Y
equivalent. The effective fields R = -a~/aR are:
Although these look very different it is easy to verify that
they yield the same torque:
Throughout this thesis energies will be written as E(MxtM ,MZ) Y rather than explicitly taking into account the constraint on the
length of the magnetization.
Applied Fields
The torque due to the applied fields is:
where go is the static applied field and E(Ztt) the microwave
field. The magnitude of the microwave field is much less than
Ho.
~emagnetizinq Field
The torque due to the demagnetizing effects is written in
terms of an effective demagnetizing field Hd:
In the special case of a uniformly magnetized ellipsoidal sample
the demagnetizing field is uniform and may be written in terms
of the demagnetizing tensor D:
The components of IJ are the demagnetizing factors for the three
principal axes of the ellipsoid. A thin disk may be treated as
the limiting case of an ellipsoid, with a demagnetizing factor
Dl, if the magnetization is in the plane of the disk, and DL if
the magnetization is perpendicular to the sample plane. If the
ratio of the sample thickness to diameter is small Dl, will be
small and DL approximately 1. For an infinite slab Dl, = 0.
Kraus and Frait[42] give an empirical expression for the
demagnetizing field at the center of a disk when the
magnetization is in the sample plane:
where R is the thickness to diameter ratio. For the samples
used in the present work R was approximately
2aMS = 3.2 kOe and MS/~,<0.4 at the fields of interest, so the
demagnetizing field was of the order of 30 Oe. This field is
included in the calculation by replacing Ho by Ho+Hd, Hd points
in the opposite direction to H, of course. For the rest of this
chapter this in plane demagnetizing field will be ignored. In
the general case where the magnetization may point out of the
sample plane the z-component of the demagnetizing field is
-4nMZ where MZ is the component of as normal to the sample plane. An important point is that only the demagnetizing field
due to the static magnetization is considered here. The
microwave demagnetizing effects are taken into account by means
of Maxwell's equations,
Maqnetocrystalline Anisotropy
As a result of spin-orbit coupling the energy of a
ferromagnet depends on the direction of the magnetization with
respect to the crystal axes. This energy is called the
magnetocrystalline anisotropy energy. The term
'magnetocrystalline anisotropy' will recur many times in this
thesis so it will be abbreviated as 'MCA'. MCA is often
referred to as, simply, the anisotropy. It is thought that this
usage could lead to confusion with the sought after anisotropy
of the FMR linewidth, hence the abbreviation.
MCA is important in this work for a number of reasons. The
large torque arising from MCA introduces experimental problems
in mounting the samples; they tend to rotate unless H, is
accurately aligned with a principal axis. The dragging of the
magnetization due to MCA can lead to problems of analysing the
data. Both MCA and magnetic damping are a consequence of
spin-orbit coupling. The observation of Furey (quoted b.y a
number of authors, see for example Kambersky [2] or Franse [22])
that a large part of the MCA of Nickel arises from the electron
states around the X, hole pockets played a part in focussing the
attention of workers interested in magnetic damping on those
parts of the Fermi surface.
The standard method70f treating MCA is to write the part of
the free energy per unit volume of the sample which depends on
the orientation of the magnetization as a series in the
direction cosines of the magnetization with respect to the cubic
crystal axes:
where K4, M2, M3 ... are the magnetocrystalline anisotropy - constants, S = afa: + aza; + a$a: and P = afa:a$, and a,, a,, a3
are the direction cosines. Any combination of the direction
cosines which has the necessary cubic symmetry can be written in
terms of S and P (see for example Aubert et aP[43]).
MCA in ~ickel at room temperature is well described by the
first two terms in this series, however the description becomes
more complicated at low temperatures where four anisotropy
constants are required plus two additional constants which are
not part of the series (2.6) (Gersdorf[44]). This will be
discussed further in chapter 4. For the calculations in this
chapter the series (2.6) with K1, K2 and K3 will be used.
In Nickel K1 is negative so the MCA energy is a minimum if
the magnetization points along a ( 1 1 1 ) direction (the 'easy
axis'), and a maximum if the magnetization points along a (100)
direction ( the 'hard axis'). There is a saddle point in the
energy about the (110) directions. The MCA torque on the
magnetization is zero for these three directions. If the
magnetization lies in either a (100) or (110) normal crystal
plane there is an in plane torque but there is no MCA torque
tending to rotate the magnetization out of that plane.
The torque on the magnetization is written in terms of the
effective field BAN:
With As along the [001] direction, as in figure 2.1, the
direction cosines are (see Figure 2.3):
I ~ i q u r e 2.2 The crystal axes in the [IT01 plane.
Figure 2.3 The angles used in determining the effective MCA
fields. The [100] and [OlOl axes are in the y-z plane. The
direction cosines are: u l = cos(6,) = (1/flMS)(My - MZ); U, = code,) = ( I / ~ M ~ M + M ~ ) :
Y U, = COS(~,) = M./M,.
Recall that the sample normal is along b 1 7 0 ~ The polynomials S
and P are:
We are interested in the case where M and MZ are small, Y
M = m MZ = m and Mx = MS. Keeping only terms of first order Y Y' Z
in m and mZ the components of the effective MCA field are: Y
Note that using this method to calculate the effective field
there is no component of BAN in the direction of 8,. The
non-zero components of HAN are proportional to the deviation of
the magnetization from equilibrium, and in equilibrium (m = O r Y
The effective MCA field when 8, is parallel to the [1101 or
[ 1 1 1 1 directions is found in a similar manner. If BAN is
written:
BAN = (0,-amy/MSt-ymZ/Ms)
then the coefficients a and y are as listed in Table 2-1. This
notation is that used by Cochran and ~einrich[37], this y should
not be confused with the gyromagnetic ratio. Note that a and y
are equal when &is is parallel to a (100) or a ( 1 1 1 ) direction
but differ when as is parallel to a (110) direction.
Exchanqe
The exchange interaction leads to an energy which depends
on the angle between electron spins. This interaction is
responsible for ferromagnetism and in a ferromagnet this energy
is a minimum if all of the spins are parallel, or, equivalently,
if the magnetization is uniform in space. Any non-uniformity of
the magnetization increases the exchange energy.
An expression for this energy increase may be found from
. symmetry arguments (see Turov[45]). For example, the energy
expression must have the symmetry of the crystal lattice and
must be invariant on replacing by -&i. The most general
expression involving OMx, OM and WZ which does not depend on Y
TABLE 2-1
Orientation a 7
of as
[ool 1 2K 1 /MS 2~ 1 /MS
[ 1 1 1 1 -4/3 (K1 /MS+K2/3Ms+2K3/3Ms) -4/3 (K1 /MS+K2/3Ms+2K3/3MS)
[1101 K ~ / M ~ + K ~ / ~ M ~ + K ~ / ~ M ~ -(2K1/MS+K3/Ms)
the direction of a or the direction of the gradients of the components of is, to second order in and V:
where A is the exchange constant, A = 1 x 1 0 - ~ erg/cm in Nickel,
see Table 4-1. It can be shown[411 that the effective exchange
field is:
B exc = 2(A/M:) (V'M~,V~M Y ,V2MZ)
With a spatial dependence of exp(ikz) this expression becomes:
The exchange torque may be important in ferromagnetic
metals because the limited penetration of microwaves leads to a
spatial variation of the magnetization. The magnitude of the
exchange field is discussed below. It is a small effect in
Nickel.
Maqnetostriction
If the sample is strained in any way there is a torque on
the magnetization due to magnetostriction. The sample may be
strained during preparation, for example, by mechanical
polishing followed by inadequate electropolishing, or it may be
strained during an experiment by the mounting used to hold the
sample. If the sample is soldered to a diaphragm of a different
metal the differential thermal contraction on cooling may lead
to strain. A uniform stress shifts the position of the
resonance; shifts up to 100 Oe can easily be produced (Cochran
and Heinrich[37]). A non-uniform stress broadens the resonance
as different parts of the sample resonate at different values of
the applied field. Care was taken to avoid straining the
samples in the present experiments and therefore
magnetostriction is not included in the calculation of
linewidths and line positions. The effect of a uniform stress
on the field at which FMR occurs is discussed by ~ac~onald[46].
Magnetic relaxation processes are included in a calculation
of FMR by introducing a damping torque. Two forms for this
damping torque are in common use, the Gilbert form:
where G is the Gilbert damping parameter, and the
Landau-Lifshitz form:
where A is the Landau-Lifshitz damping parameter. The field
entering the Landau-Lifshitz form is the effective field,
including the applied, demagnetizing, MCA and exchange fields.
In both forms aWat is perpendicular to R, that is the length of
the magnetization remains constant during relaxation back to
equilibrium. It can be demonstrated (see for example
Baartmani471) that the two forms are equivalent but that the
values of the damping parameter and the gyromagnetic ratio
deduced from experimental data will be slightly different
depending on whether the Gilbert or the Landau-Lifshitz form is
used in the analysis of the data. If G and -y are values
appropriate for Gilbert damping, the corresponding parameters
for Landau-Lifshitz damping are:
For light damping, (G/yMs) << 1 , the two forms are equivalent
and the parameters have the same values. In Nickel
G/~M, = 0.026 at room temperature and is of the order of 0.15 at
4.2 K (based on the value of G required to reproduce the
experimental linewidth assuming a wavenumber independent
damping). Thus there is no essential difference between the two
forms of damping for Nickel in the temperature regions in which
we are interested. The Gilbert form will be used in this work.
Calculation of the Permeability
Gathering these effective fields (2.3, 2.8, 2.9) into the
Landau-Lifshitz equation and using the Gilbert form for the
damping torque we have:
where Seff = ff, + R~ + aexc Recall that a = Rs + s(?,t). as is parallel to the x-axis, the z-axis points into the slab,
parallel to the sample normal. i% and 5 are assumed to vary as
exp(i(kz-at)) and are considered to be small so that quantities
second order in m and h may be neglected. Writing out the three
equations (2.12):
m, = 0
o 2A o G -i-m + (H,+y+-kz-i- Y Y Ms
)mz = MShz YYMS
Solving for 6 in terms of 6 gives the susceptibility tensor 3,
where 6 = 2.6 or:
where:
It is interesting to note that the damping torque appears in the
magnetic field term for Gilbert damping. Had we used the
Landau-Lifshitz form for the damping torque the second and third
equations of (2.13) would have read:
The damping enters the susceptibility through the frequency.
The microwave demagnetizing field, due to m,, may now be
included. Since V-5 = 0 from Maxwell's equations, we have
b, = h, + 4 m z = 0 or hZ = -4rrmz. Combining this with equations
( 2 . 1 3 ) we may solve for my, mZ, and h, in terms of h the Y'
quantity which will be related to the applied microwave field.
with B = H + 4uMs, Y Y
BZ = HZ + 4aMs and H and HZ are given by Y
equation ( 2 .15 ) . The ratio m /h defines what will be called Y Y
the effective susceptibility X . This is NOT a component of the
susceptibility tensor. The ratio mz/m indicates the degree of Y
ellipticity#of the precession of the magnetization.
The ratio b /h is given by the effective permeability Y Y
where Bo = HI + 4rMs. Ignoring MCA, exchange and damping the
permeability becomes:
There are two interesting values of magnetic field in this
expression. When Bo = ( 0 / y ) , that i s Ho = - 4nMs, the permeability vanishes, the microwave 5 field is zero. This
condition corresponds to ferromagnetic anti-resonance ( F M A R ) .
When B o H o = the denominator vanishes and the permeability
becomes very large. This condition is ferromagnetic resonance.
The effective permeability contains all the information
required for this calculation. However when we consider
dragging of the magnetization in Section 2.3 it will be
impossible to define a single quantity analogous to the
effective permeability. It will be necessary to work with the
permeability tensor. For comparison with the calculation of
Section 2.3 it is useful to write out the permeability tensor
Maxwell's Equations
We now have the microwave permeability of the magnetic
metal. This permeability is available in two forms, the
permeability tensor (2.20) in which the microwave demagnetizing
field has not been included, and the scalar permeability (2.18)
which relates b to h with the microwave demagnetizing field Y Y
explicitly taken into account. The problem now is to combine
the permeability (either (2.20) or ( 2 . 1 8 ) ) with Maxwell's
equations to solve the boundary value problem of the reflection
of microwaves from the metal surface. Maxwell's equations are,
in CGS units:
v-l5 = 0
In this section we treat the case where the conduction electron
mean free path is much smaller than the microwave skin depth.
Ohm's law is valid so that the current density 5 is related to the electric field G by:
where oo is the dc conductivity. With the space and time
variation exp(i(E.2-at)) the two curl equations become:
where e is the dielectric constant of the metal. The
displacement current term in Ampere's law, i(we/c)Z, may be
neglected at microwave frequencies. For metals oo is typically
lo1' sec-l and e is of order 1 so that for w 1011 sec-',
4noo >> oe. For propagation along the z-direction (E = kg)
these equations become:
'xx 'xy pxzl /hx
The microwave demagnetizing condition, bZ = 0, is included
automatically in the third of equations (2.24). Eliminating the
electric field leaves the three equations:
from which one gets the equations:
where 6' = c2/2nooo, 6 is the skin depth for a permeability
p = 1. This homogeneous system of equations has a so1u;ion only
if the determinant of the coefficients is zero. The condition
that the determinant of the coefficients equal zero determines
the values of the wavevector k of the microwave fields in the
metal.
For the case we are considering it is not necessary to work
with this tensor representation. From (2.20) and (2.26) we
have :
bx = hx
bx = -i k26'/2 hx
so that:
-ik26'/2 = 1
From (2.18) and (2.26) we have:
by = Yhy
by = -ik26'/2 h Y
so that:
Consider first the situation with no exchange ( A = 0). The
permeability is then independent of k2 and there are two
solutions for k2:
These correspond to four waves which can propagate in the metal,
two of which propagate in the +z direction and two of which
propagate in the -2 direction. Only the waves travelling in the
+z direction will be considered here because we deal only with
the case where the slab thickness is much greater than the skin
depth.
The wave described by k = I/zr~<6 corresponds to the result
expected for a non-magnetic metal. This wave is linearly
polarized with its fi field parallel to MS (the x-direction).
The wave described by k = is the interesting one, as the
wavevector exhibits the resonant behaviour of the permeability.
The microwave fi field is perpendicular to as and has both y and z components. The fi vector traces out an elliptical path in the
opposite sense to the precessing components of the
magnetization, so that hZ = -47rmZ. The ratio h,/h was given Y
above in equation (2.18)- The electric field has only x
components and the 5 field has only y components as expected.
If exchange is included in the calculation the permeability
depends on k2 and the relation r + i k26' = 0 becomes a cubic
equation in k2:
where H' = Ho+o-i ( w / y ) (G/7MS) , HZ = Ho+y-i (W /Y ) (G/7MS) and Y
B' = H' + 4MS. There are four values of k 2 and eight waves
which can propagate in the metal. Consider only those waves
which propagate in the +i direction. One value ~f k is
k = 42i/6 and corresponds to the non-magnetic wave as above.
The other three waves have their fi fields perpendicular to MS.
For values of the applied magnetic field far from the resonance
field value these three waves can be categorized as one having
primarily electromagnetic character, corresponding to the
no-exchange wave, and two which are primarily of spin-wave
character. Near FMR it is not possible to make this
distinction. For a detailed discussion of the nature of these
waves see Cochran et a1[48l.
The Boundary Value Problem
Having found the wavevectors, k, of the waves in the metal,
the boundary value problem may now be solved. As a reminder,
the geometry and field amplitudes are indicated in Figure 2.4.
Microwaves are incident normally on the surface of a metal slab,
the incoming microwaves, ei, hi, being linearly polarized with
the a field along the x direction, parallel to the applied field ,a, and the magnetization as, and the E field along the
y direction. We wish to determine the reflected field
amplitudes, err hr, and em, hm the field amplitudes transmitted
into the metal, as. well as the reflected and the absorbed power.
The fields and li must satisfy the boundary condition that
their tangential components be continuous across the interface.
As an example consider the case with no exchange. With the
microwaves incident as in Figure 2.4 only the resonant wave is
excited in the metal. The boundary conditions are:
From Maxwell's equations, in the vacuum:
Figure 2 . 4 Geometry for the boundary value.problem.
and in the metal:
where:
ZS is the surface impedance of the metal, and the second
equality follows from equation (2.30). For a non-magnetic metal
having a conductivity equal to that of Nickel at room
temperature ( p = 7.2~10'' Ocm, o, = 1.25 x 10" sec-', see
Table 4 - 1 1 and a microwave frequency of 24 GHz, one obtains
Zs * 2.2~10-~(1 - i ) .
Solving equations (2.32) for hr and hm we find:
The reflected power, Pr, is found using the Poynting vector
Since 6 is perpendicular to E and erx = -h the reflected power rY
is proportional to lh 1 2 . The ratio of the reflected power tO rY
the incident power, Pi, is:
The absorbed power, Pa, is the incident power less the reflected
power :
Writing Zs = r + ix where both r and x are small, and ignoring
quantities of second order:
Therefore, for small Zs (the usual situation):
From (2.38) and the definition of Zs, the absorbed power is
proportional to the real part of r/--irT;.
Returning to the case in which exchange is included, three
magnetic waves will be excited. We have two boundary conditions
for the tangential components of and k (2.32): however there
are four unknown amplitudes (one reflected wave and three
transmitted waves in the metal). Two additional boundary
conditions are therefore required. These are the spin-pinning
conditions on the amplitude of the components of the
magnetization and the spatial derivatives of the magnetization
at the surface, see Rado and Weertman[50]. For a uniaxial
surface anisotropy energy, E = KScosZ(Bf, where 6 is the angle
between the magnetization and the x-axis and KS is the surface
anisotropy constant, with the axis parallel to the equilibrium
direction of the magnetization the boundary conditions are:
the subscript 0 indicating that the quantities are evaluated at
the surface of the slab, z = 0. These boundary conditions have
been discussed by Cochran, Heinrich and Dewar[48]. If Ks = 0
the conditions (2.39) become aiii/azl, = 0 corresponding to
'unpinned' or free spins. If Ks is large we have % l o = 0
correspo'nding to spins pinned at the surface.
With the geometry of Figure 2.4, and using space and time
variations exp(i(kz-at)), the four boundary conditions (2.32)
and (2.39) become:
The subscripts 1 , 2, 3 refer to the value of k for the three
roots of equation (2.31). The electric field components in the
metal are related to the 8 components by (see equations (2.33)
and (2.34)):
where:
e = Z jx sj ?Y
Z = -ioazkj/2c !Tj
The components of the microwave magnetizations ii are related to j
the h by equations (2.17) with the appropriate value of k: jy
Combining equations (2.401, (2.41), (2.42) we may solve for the
field amplitudes hr, h,, h, and h,. no wing the reflected wave
amplitude enables one to calculate the power absorbed by the
sample (see equation 2.37). The expressions for the wave
amplitudes are complicated and there is little point in writing
the equations out in detail. However a computer program has
been written to calculate the susceptibility and to solve the
boundary value problem numerically. The program calculates the
absorption and the absorption derivative as a function of the
applied field for a given set of parameters which characterize
the metal.
Application to Nickel
Calculated curves of the absorption and the absorption
derivative as a function of the applied field are shown in
Figure 2.5. Parameters appropriate to Nickel at room
temperature and a microwave frequency of 24 GHz were used in the
calculation. The values of the parameters are listed in
Table 4-1 in Chapter 4. The field at which resonance occurs,
Hfmr and the FMR linewidth, AH, are indicated on the figure.
A brief discussion of the effect of the various parameters
on the FMR absorption will be given here. This discussion will
be amplified in Chapter 4. In the absence of MCA, damping and
exchange, FMR occurs at the applied field where the permeability
becomes infinite:
(see equation (2.18)). For the remainder of this section this
field value will be termed HfmrO. Including MCA, but neglecting
damping and exchange, the permeability becomes, for the applied
field along a (100) direction and the sample normal along [ 1701:
The permeability becomes infinite when:
(Ho+2K1/MS)(Ho+4~M~2K1/~ = S
ABSORPTION
(ARB. UNITS )
A P P L I E D F I E L D (Me)
ABSORPTION
D E R I V A T I V E
(ARB. UNITS )
A P P L I E D F I E L D (Me)
Fiqure 2.5 Calculated absorption and absorption derivative.
Parameters appropriate for Nickel at room temperature were used
in the calculation, see Table 4-1. The microwave frequency was
23.95 GHz.
that is the resonance field is shifted from HfmrO by 21K1 l/Ms.
The shift is to a higher field if K1 is negative, as it is for
Nickel. For MS along a ( 1 1 1 ) direction the shift is to a lower
field for Nickel (see Table 2-11. The direction of the shift
for Ms along a (110) direction is not obvious as the anisotropy
fields a and y are different. The value 21Kll/~~ provides a
measure of the magnitude of MCA effects. For Nickel at room
temperature 2 1 ~ l l / ~ ~ 240 Oe, and at 4.2 K 21K11/MS 0; 4.92 kOe.
Including damping, but neglecting MCA and exchange, the
permeability becomes:
. where the second order term in (G/yMS) has been neglected in the
second equation. The permeability is now complex. It is
important to remember that in an experiment we measure the
absorbed power which varies as the real part of the square root
. of - i times the permeability, see equation (2.38). The real and
imaginary parts of the permeability become mixed in taking the
square root. The result of damping is a shift in Hfmr to a
higher field; and a broadening proportional to, and of the order
of, (o/y)(G/yMS). At a frequency of 24 GHZ, at room
temperature, (o/y)(G/yMS) is approximately 200 Oe, while at
4.2 K it is approximately 1100 Oe.
Including exchange, but neglecting MCA and damping, the
permeability becomes:
The wavenumber, k, depends on the field through the
permeability:
Again the permeability is complex. Since k2 is proportional to
1 / a 2 the exchange field 2 ~ k ' / ~ ~ is proportional to (~/6') or
(~0,) where oo is the dc conductivity. A large conductivity
results in a small skin depth and large gradients of the
magnetization, i.e. the exchange field will be large. For
Nickel at room temperature and a frequency of 24 GHz the skin
depth 6 = 0.9 Dm. At FMR lei = 20 so k 5 x 10' cm-l, and
2Ak2/~, = 10 Oe. The exchange field shifts the resonance. Some - broadening is also produced as the field distribution in the
metal is not described by a single wavenumber.
An idea of the relative importance of the damping and the
exchange contributions to the shift in FMR and the linewidth may
be had from the numbers listed in Table 2-2. Values of the
TABLE 2-2
Hfmr (kOe) 6H(Oe) M(0e)
G=O, A=O 5.326 -- -- G=2.45~10~sec'~, A=O 5.337 1 1 300
G=O, A=l.O~lO-~erg/cm 5.290 -36 50
G=2.45~1O~sec-~ 5.314 -12 320
A=l.O~lO'~erg/crn
Calculated values for the resonance field, Him,, the shift in
peak position, 6H = Hfmr - Hfmr~, and the linewidth, AH. Parameters appropriate for Nickel at room temperature were used
in the calculations: 4xMs=6.16 kOe, f=23.95 GHz, p=7.2x10q6 Qcm,
The MCA constants have been set equal to zero for these
calculations.
resonance field, Hfmrt the shift 6H = Hfmr-HfmrOt and the
linewidth, AH are listed for four situations: (i) no damping, no
and (iv) damping and exchange. Room temperature Nickel
parameters were used in the calculations. Since MCA does not
contribute to the linewidth it has not been included in these
cabculations. As can be seen from Table 2-2 the shift due to
- exchange is approximately three times that due to damping, and
in the opposite direction, while the linewidth is dominated by
'the damping. Of course the shifts for case (iv) are not just
the sum of those for cases (ii9 and (iii9.
2.3 Arbitrary Orientation of the Magnetization
The calculation described in Section 2.2 is valid only for
the very stringent conditions that the sample plane coincides
with a (110) crystal plane, that the applied field is parallel
to the sample plane, and that the applied field be parallel to
one of the three principal crystal axes (100), (!lo), or. (1119,
in that plane. Since there is no static MCA torque on the
magnetization for these directions the magnetization will be
parallel to the applied field, at least at the field values of
interest. This is the simplest geometry to treat and is that
aimed for in an experiment.
Deviation from this-ideal situation may occur for a number
of reasons. The sample plane may not coincide with a ( 1 10 )
plane. The applied field may not be exactly aligned with the
crystal axis, being tilted out of the sample plane or rotated in
the plane. In such a case the magnetization will not, in
general, be parallel to the applied field. A calculation of the
FMR absorption must take this into account. The result of the
lack of alignment is a shift and a broadening of the absorption
line relative to the position and linewidth which would be
expected if those effects were not considered. The magnitude of
these so-called dragging effects in Nickel is greatest at low
temperatures where the MCA becomes large.
The motives for carrying out the following calculation are
twofold. First to determine the effect of a small misalignment
- % I -
of the field with the crystal axes, either in or out of the
sample plane, on the absorption. Second, to obtain an idea of
the angular variation of the resonance field, Hfmr, in the (110)
plane. The calculated variation of the resonance field may be
compared with the experimental variation to determine the
location of the principal axes in the sample plane. The angular
variation of the resonance field has a maximum when the.applied
field is parallel to a (100) or (110) axis in the sample plane.
It is straightforward to locate these axes in an experiment by
rotating the magnet and finding the angle for which the
resonance field has the largest value. The.angular variation
has a minimum near the ( 1 1 1 ) axis but the exact position of the
minimum depends on the value of the MCA constants. To determine
the location of the ( 1 1 1 ) a.x.is it is necessary to compare the
calculated and the experimental angular dependences. Also the
agreement between the calculated and experimental angular
variations serves as a test of how closely the sample plane
coincides with a (110) crystal plane.
Three steps are involved in the calculation of the
absorption as a function of the applied field. First the
orientation of the magnetization in equilibrium for a given
magnitude and orientation of the applied field must be
determined. The permeability is then found using the
Landau-Lifshitz equation. Finally the boundary value problem is
solved. The second and third steps are the same as in the
calculation outlined in Section 2.2 but the algebra is much more
involved.
A computer program was written to carry out this
calculation numerically. The program yielded values for the
orientation of the magnetization, the permeability, the
wavevectors in the metal, the absorption and the absorption
derivative. In the calculation that was programmed it was
assumed that the sample plane was a (110) crystal plane6 The
applied field was allowed an arbitrary orientation with respect
to the sample plane and the crystal axes. A local conductivity
was assumed and exchange was neglected because of the
complications involved. As was seen above exchange effects are
comparatively small in Nickel. A further assumption, implicit
in the calculation of Section 2.2, is that the sample forms a
single domain. The magnetization changes by rotation only.
The geometry assumed is shown in Figure 2.6. The x-y-z
axes are the same as those defined in Figure 2.1: the z-axis
pointing into the slab, parallel to the sample normal, and the
y-axis being parallel to the incident microwave magnetic field.
This coordinate system will be called the 'laboratory frame'.
The orientation of the applied field and the magnetization are
specified by angles ( OH, and (O,#) respectively, the
equilibrium values of (Or#) being (OM,#M). The sample normal is
the [ti01 axis. The sample may be rotated about the [li01 axis,
the angle between the [0011 axis and the x-axis being $.
In the calculation of the permeability it is desirable to
work in a coordinate system with one axis parallel to the
Figure 2.6 The angles required for the calculation of
Section 2.3. The x-y-z axes define the laboratory frame where
the sample lies in the x-y plane, the z-axis points into the
slab and the y-axis is the direction of the microwave magnetic
field. (a) The applied field, a,. (b) The magnetization, as,
and the magnetization frame x'-y'-z'. The x' axis is parallel
to as, the y' axis lies in the x-y plane. (c) The crystal axes,
@ is the angle between the [ 0 0 1 ] axis and the x-axis. The
[ 0 0 1 ] , [ 1 1 1 ] and [ 1 1 0 ] axes lie in the x-y plane.
magnetization as the Landau-~ifshitz equation has a simple form
in such a system. The 'magnetization frame', (x',y',z'), is
defined with the x'-axis along Ms. A convenient choice for a
second axis is to have the y'-axis in the sample plane. The
magnetization frame coincides with the laboratory frame if the
magnetization lies along the x-axis.
If li is a vector in the laboratory frame, and li' is the
same vector in the magnetization frame then:
where T is the matrix representing a rotation about the z-axis
by mMr followed by a rotation about the y'-axis by (a/2 - OM).
L J
The permeability tensor is calculated in the magnetization frame
2' . The transformation to the lab frame is:
k =T- l j ' p p
where T-l is just the transpose of T.
-75-
Equilibrium Orientation of the Maqnetization
Three equivalent ways of stating the equilibrium condition
for the orientation of the magnetization are that Ms is parallel
to the effective static internal field, that the torque on the
magnetization is zero, or that the free energy is a minimum. In
the calculation that was programmed the minimum of the free
energy was found.
The contributions to the free energy are due to the applied
field, E = -e-RQ, the demagnetizing field, E =47M:, where the
in-plane demagnetizing field is neglected (see Section 2.21 , and
MCA, the energy being given by the series (2.6). The total free
energy is:
This expression written out in terms of OH, QH, $, 0 and Q is
very complicated. No apology is offered for not including it
here. It is straightforward to set up a numerical procedure to
find the angles OM and QM which minimize this energy.
To demonstrate the magnitude of this dragging effect plots
of the calculated variation of OM and @M with the applied field
are shown in Figures 2.7 and 2,8. MCA constants for Nickel at
4.2 K were used in the calculations. These are listed in
Table 4-1 in Chapter 4. For the calculatisns shown in
Figure 2.7 the applied field was parallel to the sample plane,
Figure 2.7 Calculated variation of the the direction of the
magnetization with the applied field. The applied field was
assumed to lie in the sample plane and parallel to (a) [ 1 1 1 ] .
magnetization and the [ 0 0 1 ] axis is plotted. For (b). (c) and
(dl the direction of the applied field OH) is indicated by a
dashed line. Parameters appropriate for ~ickel'at 4.2 K were
used in the calculation, see Table 4-1. The MCA fields I K I I / M ~ and 21Kll/MS are indicated on t,he figure.
5 10 APPLIED FIELD (kOe)
Figure 2.8 As Figure 2.7 however the applied field was assumed
to point 5' out of the sample plane with its in-plane projection
parallel to [0011. The angle between the in-plane components of
a, and B,, (@M-@H), and the out-of-plane angle, (OM-OH), are
plotted. Parameters appropriate for Nickel at 4.2 K were used
in the calculation, see Table 4-1.
-78-
and so the magnetization was also parallel to the plane. The
angle between the magnetization and the [001] direction,
( 4~ - $1, is plotted rather than mM as this permits several plots to be displayed on one figure. The angle the field makes
with the [001] direction, (mH - $1, is shown by a dashed line.
Curves are shown for the field along (a) [111]; (b) along [110];
(c) along [001]; and (d) 16' from the [001] direction. This
last direction exhibits the most spectacular dragging effects.
In curve (a) the magnetization is parallel to the field for all
values of the field. For curves (b) and (c) the magnetization
is parallel to the field for fields above approximately IKII/MS
(2.46 kOe) and 21K11/MS (4.92 kOe) respectively. Alignment of
the magnetization with the field occurs at exactly these field
values if K2 and K3 are zero. Ferromagnetic resonance in Nickel
at 24 GHz at 4.2 K occurs at field values of 5.5 kOe and 10 kOe
respectively for these two directions so that dragging does not
affect the observed resonance lines when the external field is
applied along any of the three principle axes in the sample
plane. In curve (dl it should be noted that even when the
magnitude of the applied field is 15 kOe there is an angle of 5*
between the magnetization and the field.
For the calculation shown in Figure 2.8 the applied field
was tipped 5' out of the sample plane but with its in plane
projection parallel to the [001] axis. The variation of mM is similar to that of Figure 2.7 ( c ) . The variation of OM is
approximately linear in field with a kink at H, = 2IK1 I/M,, the
field at which OM becomes zero. The angle between &fS and So
decreases from 1.5' at H, = 21~ll/M~ to 0.5' at Ho = 12 kOe.
The permeability
The Landau-Lifshitz equation, rewritten for reference, is:
where the primes indicate quantities measured in the
magnetization frame, since it is convenient to carry out the
calculations in that frame. The effective fields (both static
and dynamic) are found by taking the derivative of the free
energy with respect to t-he magnetization:
where the energy E is given by equation (2.491, and we use the
notation Ex to denote the derivative of the free energy with
respect to M; evaluated at equilibrium (Mi = Ms, M' = 0, Y
MI = 0). H; is the static internal field. Expanding H' for Y
small deviations from equilibrium (Z ' ) :
Similarly:
H~ = -a2~/am;210rn; - a2~/am;am;l,m' Y (2.52)
The cross derivative could be made equal to zero by an
appropriate rotation of the magnetization frame about the
x'-axis. However with the frame as defined it is necessary to
carry this term through. The term is zero if the magnetization
lies in the sample plane or in a (100) or (110) crystal plane.
The second derivatives have two parts, one due to MCA, and
.one due to the microwave demagnetizing field. It was seen above
(equation (2.24)) that the microwave demagnetizing field was
treated automatically by Maxwell's equations without including
it in the calculation of the susceptibility (2.13). By
reviewing the steps outlined in Section 2.2 it can be seen that
had the microwave demagnetizing field been included in the
calculation of the susceptibility it would have been counted
twice in the complete calculation. The effective fields (2.51
and 2.52) must include only the MCA contribution. With:
then:
If Ms lies in the sample plane (OM = n/2) and along a principal
axis, then E = MSa and EZ, = Msy where a and y are the YY
effective MCA fields listed in Table 2-1. The equations of
motion become (with a time variation exp(-iot)):
The permeability is found following the same steps as in
Section 2.1. The non-zero components of the permeability tensor
are:
This permeability tensor is transformed into the lab frame:
In general all nine components of 2 are non-zero.
Maxwell's Equations
The permeability tensor, 2, is combined with Maxwell's
equations in the same way as above (see equation (2.26)). We
are neglecting exchange and therefore there are two values of k 2
corresponding to two forward and two backward propagating waves
in the metal. If the magnetization is parallel to the sample
plane these are the non-resonant wave with FI parallel to
as, and the resonant wave with E perpendicular to as. If the magnetization is perpendicular to the sample plane the two waves
- correspond to circularly polarized waves, one of which is
resonant and the other is not. In the general case both waves
have some resonant character.
The Boundary Value Problem
Both waves in the metal will be excited unless the
magnetization is parallel to either the x or y-axes. As a
result the reflected microwaves are elliptically polarized. The
boundary conditions are (see Figure 2.4):
where el,h, and el,h2 are the amplitudes of the two waves in the
metal. From these equations, the relations between e and h j j '
e = Z . h j SI j r
and the equations (2.26) with the appropriate value
of k, the unknown field amplitudes hrx, h ry' hlxt hly' hzx and
h 2 ~ may be found. The reflected power is:
The calculation follows through as in Section 2.2.
An Example of the Effects of Draqginq
The results of two calculations of the absorbed power as a
function of the applied field are shown in Figure 2.9. The
parameters used in the calculations were appropriate for Nickel
at 4.2 K. These are listed in Table 4-1. For both calculations
the applied field was parallel to the sample plane and 16' from
the [001] direction (see Figure 2.7 for the variation of eM with the applied field for this situation). Curve (a) is the result
that would be expected if dragging were not considered, if the
magnetization remained parallel to the applied field for all
values of the field. This is not what happens of course. In
curve (b) the dragging has been taken into consideration. The
shift in FMR to a lower field and the line broadening are clear
from the comparison of the two curves.
Although graphs of the angular variation of Hfmr and the
linewidth, AH, are presented in Chapter 4, a few numbers are
worth quoting here. The linewidth is approximately 1600 Oe at
4.2 K. If the applied field is 1 " away from the [001] axis, in
the sample plane, the calculated linewidth is 10 Oe greater than
the [0011 linewidth. A 2' misalignment results in an additional
broadening of 40 Oe. If the applied field is tipped 2' out of
the sample plane with its in plane projection parallel to the
[001] axis the additional broadening is 10 Oe (compare the
angular variation of By and eM in Figure 2.8). These
differences are of the order of the experimental uncertainty.
ABSORPTION
(ARB. UNITS )
I 1 5 10
APPLIED FIELD (We)
Figure 2.9 Calculated absorption in Nickel at 4.2 K, with the
applied field assumed to lie parallel to the sample plane and
16" from the I0011 axis. (a) the magnetization was assumed to
remain parallel to the applied field for all values of the
applied field. (b) the lack of alignment between the
magnetization and the applied field was considered. Parameters
used are listed in Table 4-1.
A Further Point
If we assume that the magnetization and the applied field
are parallel then, by analogy with (2.45), resonance will occur
at the value of the applied field which satisfies:
where # is the angle between the applied field and the [001]
axis: with $ = 0, 4 = mH = The effective MCA fields are
(compare '2.53) :
If we consider K1, K2 and K3 these effective fields are, in a
(110) normal crystal plane:
If 21Kll/Ms << Hfmr then Ms will be parallel to the applied
field at the fields at which FMR occurs and the expression
(2.55) will describe the angular variation of the resonance
field (neglecting the damping and exchange shifts). This is the
situation in Nickel at room temperature where 21Kll/Ms = 240 Oe
and Hfmr= 5 kOe. If 21Kll/Ms * Idfmr as in Nickel at low
temperatures where 2]Kll/Ms = 4.92 kOe, dragging will have a
large effect and (2.55) will not describe the angular variation
Of Hfmr*
The expression (2.55) combined with the variation of the
angle between the magnetization and the applied field, @M - #H, enables one to give a simple picture of why dragging leads to
shifts of FMR and to broadened lines. In Figure 2.10 curves (c)
and (d) of Figure 2.7 are plotted together with Hfmr(@) of
(2.55) plotted as @ against Hfmr. The microwave frequency in
Figure 2.10(a) is 24GHz and 9.5GHz in ~igure2.l0(b). If the
magnetization were parallel to the applied field FMR would occur
at the intersection of the two curves @H and Hfmr($J), for
example at the points A and B in (a). As a first approximation
we may assume that the effective static internal field (parallel
to the magnetization) is equal to the applied field. This is
true only if the magnetization is parallel to the applied field,
Fiqure 2.10 Calculated variation of the direction of the
9Q -
60 -
magnetization with the applied field, as in Figure 2.7. The
- -. \
./ 1
/)' / @ @
(b)
short dashed lines are a plot of Hfmr(#) from eqn (2.57) at
- -eC / @ f l D --
- C & -
-30 ---- 1
0 I
5 10 APPLIED FIELD (kOe)
(a) 24 GHz, (b) 9.5 GHz, plotted with the angle on the vertical
axis. FMR occurs at the intersection of the curves #M(~) and H
fmr ( 4 ) : points A , B, D and F if dragging is ignored. points A ,
C. D and E if dragging is considered. Parameters appropriate
for Nickel at 4.2 K were used in the calculations.
but this assumption is adequate for the present qualitative
discussion. With this assumption FMR would occur at the
intersection of #M(H) and Hfmr(#), that is at the points A and C
in (a). We see that the intersection at C is at a much lower
value of the field than the intersection at B. This is the
shift in the resonance which is seen in the calculated
absorption curves shown in Figure 2.9. We could imagine drawing
dashed lines parallel to Hfmr(#) at Hfmr(#)+AHO/2 where AHo
represents the linewidth with no dragging. The calculated
linewidth with dragging would be the field interval between the
intersections of mM(H) with the curves Hfmr(@)+AHo/2. For the
[001] curve this interval is just AHo. However for the applied
field 16' off [001] this interval would be much greater than
AH0 0
The effects of dragging become more pronounced at low
microwave frequencies because of the smaller values of Hfmr. The
variation of Hfmr(#) at 9.5 GHz is shown in FigureZ.lO(b) with
the same curves of I$~(H) as in (a). If the applied field is
parallel to [001] the curve @ M ( H ) intersects Hfmr(#) at two
points, D and E. This means that the absorption line will have
two peaks, the peak corresponding to D being the peak expected
if there were no dragging and the peak at E being purely a
result of dragging. This behaviour is observed in Nickel at
4.2 K, see Chapter 4. For the applied field 16' off the 10011
axis the resonance would be expected at F if there were no
dragging, but as can be seen from the figure, no resonance will
be observed as the curves g5M(~) and Hfmr(g5) do not intersect.
This behaviour is also observed experimentally, the resonance
disappearing for small angles between the applied field and the
[0011 axis.
2.4 The Anomalous Skin Effect and FMR
The calculations outlined in Sections 2.2 and 2.3 describe
the experimental results well when the temperature is large
enough that the electrical conduction at microwave frequencies
can be described by Ohm's law using the dc conductivity. At low
temperatures the conductivity increases and the conduction
electron mean free path, I , may become comparable to the
microwave skin depth, 6. When this occurs Ohm's law does not
provide a satisfactory description of the electrical conduction,
The electrical conductivity becomes wavenumber dependent or
non-local. According to the analysis of Korenman and
Prange[3,4] the magnetic damping also becomes wavenumber
dependent at low temperatures. The wavenumber dependence of the
damping is related to the increase in the conduction electron
mean free path as discussed in Chapter 1. In this Section we
outline a procedure for calculating the FMR absorption with
wavenumber dependent quantities.
First let us consider the criteria for determining when
wavenumber dependent effects will be important. The electrons
in a metal which contribute to the electrical conduction travel
at the Fermi velocity vF. The conductivity is limited by the
scattering of these conduction electrons by phonons or
impurities. If the average time between scattering of an
electron is T, the electron mean free path is I = vFre
Ohm's law is:
where j is the current density, e is the electric field and o,
is the dc conductivity. Ohm's law states that the current
density, at a point in space at a certain time is related to the
electric field at that point and at that time only. The
relationship between the-current density and the electric field
is said to be a local one. This relation holds if (i) the mean
free path I is much shorter than the length of spatial
variations of the electric field, or ql << 1 where q is a
typical wavevector of the electric field; and (ii) the
relaxation time T is much shorter than the period of oscillation
of the electric field, or 07 << 1 where o is the angular
frequency of the electric field. These two conditions are
equivalent to saying that the electron experiences a constant
electric field between scattering events. If either of these
conditions does not hold the simple local relation between the
current density and the electric field must be replaced by a '
non-local relation which accounts for the fact that the current
density at a point at a given time depends on the value sf the
electric field at other points in space and at earlier times.
The non-local relation is written:
The electrical conductivity is described by the quantity K. If
the electric field varies in space and time as exp(i(6-%at)),
i.e. if the field is described by a single wavenumber q, and a
single frequency o, this relation becomes, for an infinite
medium:
where o(G,w) is the frequency and wavenumber dependent
conductivity, o(G,o) is the Fourier transform of ~(i,t). This
expression resembles Ohm's law and may be called a generalized
Ohm's law.
If we assume that the local Ohm's law is valid and consider
what happens when an electromagnetic wave of frequency w is
incident on a non-magnetic metal(p = I ) , the electric field in
the metal will be:
where k = ( 1 + i)/6 and 6 is the skin .depth for a local
conductivity, S2 = c2/2ro&. The electric field oscillates in
space with a wavelength 2n6 and the amplitude decays
exponentially with distance into the metal with a decay
constant 6. The wavenumber spectrum of this electric field
distribution has a maximum at q = 1/6, that is, the scale of
spatial variation of the electric field is determined by the
skin depth. The condition ql << 1 is equivalent to 1/6 << 1 .
The wavenumber dependence of the conductivity becomes important
when ql = 1 , or when the electron mean free path becomes
comparable to the skin depth expected from a calculation based
on a local conductivity.
In a magnetic metal k = d=/6 '(equation 2.291, the
effective skin depth is reduced by the permeability. Since the
permeability is large at FMR the values of q are also large and
~ ( F M R ) ~ may be >> ql for a non-magnetic metal having the same
dc conductivity. This means that wavenumber dependent effects
may be important at FMR at much higher temperatures than for a
non-magnetic metal having the same dc conductivity.
A consequence of the wavenumber dependence of the
conductivity is that the microwave penetration depth is no
longer 6 when ql > 1 but saturates at a constant value
6~ = (6'1 ) ' 1 3 [ 8 ] . For FMR measurements this means that the
effects of exchange at low temperatures are reduced over those
which would be expected if a local conductivity were used.
Some representative values for ~ickel are given in
Table 2-3 for room temperature, and for resistivity ratios of
10, corresponding to 77 R, 38, corresponding to 4.2 K for the
samples used in our experiments, and 100. Listed in the Table
are the values of the skin depth for a permeability of 1 , the
magnitude of the permeability at FMR calculated using the
program of Section 2.3 with the values of the damping parameter
required by experiment (see Chapter 4 ) , the values of the
average wavevector q for permeability 1 and at FMR assuming
q = 1/6, the electron mean free path and the dimensionless
ratios UT and ql. These numbers should be viewed as
approximate, not absolute. As can be seen U T is estimated to be
much less than 1 over the range of temperature and purity
represented in the Table. While ql is much less than 1 at room
temperature it is clear that wavenumber dependent effects are
likely to be important at low temperatures. The temperature at
which wavenumber dependent effects become important is a matter
for experiment to decide but we can see, for example, that a
local conductivity would probably be applicable for Nickel at
77 K.
Having established the need for considering a non-local
conductivity we may write down the expression for the
conductivity of a metal characterized by a spherical Fermi
surface[51 I :
In the limit q l < < l and u ~ < < l this expressi~n reduces to the dc
TABLE 2-3
77
10
10-j3
4 . ~ x I O - ~
2 . 7 ~ 1 0 - ~
3.6x104
8x108
7.6
10.0~104
250
0.25
62 = c2/2130o0 = 109p(Qcm)/4azf
~(FMR) = qdl P(FMR) 1 pzg5 = 7.2~10-~Qcrn
f = 24 GHz
v = 2.5x107 cm/sec F
I = VFT
The values of the damping parameter used in calculating the
permeability are those required by experiment, if a non-local
conductivity and a wavenumber independent damping is assumed,
see Chapter 4.
conductivity 0 , ; in the limit ql>l and w r < < l :
using arctan(x) = (i/2)ln[(l-ix)/(l+ix)]. Although the Fermi
surface of Nickel is not spherical this expression can be
expected to provide a good first approximation.
The permeability is also wavenumber dependent. The
calculation of the permeability follows exactly as in
Section 2.2. If the exchange torque is considered the
permeability depends on q. This wavenumber dependence was
treated in Section 2.2. At low temperatures the magnetic
damping becomes wavenumber dependent introducing an additional
q-dependence. This is a result of the intra-band scattering
mechanism mentioned in the Chapter 1. Following the discussion
of Cochran and Heinrich[37] the spin-flip and intra-band
contributions to the damping may be included by assuming:
where T is the temperature. The first term corresponds to the
result of Korenman and Prange[3,4] for intra-band scattering,
while the second is the result expected for spin-flip
scattering(~lliott[lO], ~ambersky[2]). The parameters a and b
are varied to match the experimental results for the temperature
dependence of the damping parameter[37]. The mean free path
which en'ters the damping, ID, is the mean free path of the
d-band electrons on the X5 hole pockets. There is no reason to
assume that this mean free path should be the same as the mean
free path which enters the conductivity. This expression
provided a good description of the FMAR results of Cochran and
Heinrich with the substitution arctan(qlD)/qlD = 1 at FMR.
Since 6 is large at FMAR, qlD is small.
The wavenumber dependent permeability, including exchange
and wavenumber dependent damping is:
A calculation involving a wavenumber dependent conductivity
requires a knowledge of how the conduction electrons scatter
after a collision with the metal surface. Two limiting cases
are usually discussed: (i) specular scattering, or mirror
reflection of electrons colliding with the surface; or
(ii) diffuse scattering, where the trajectory of an electron
after a collision with the surface is totally unrelated to the
trajectory before the collision. A complete calculation would
also have to include the effect of the applied magnetic field on
the trajectories of the conduction electrons.
Cochran and Heinrich[52] have carried out calculations of
the absorption and transmission of microwaves in ferromagnetic
materials using a non-local conductivity. Three combinations of
surface scattering, curvature of electron orbits in the applied
magnetic field and exchange were used in the calculations:
(i) specular scattering, curvature of the orbits neglected and
exchange; (ii) diffuse scattering, curved orbits and no
exchange; and (iii) diffuse scattering, curvature of the orbits
neglected and no exchange. Their results show that the field
dependence of the absorption is insensitive to the type of
surface scattering, and that neglect of the curvature of the
electron orbits has little effect. This insensitivity is a
result of the similarity of the electric field distributions in
the skin layer for the two forms of surface scattering.
The computer program we used to make calculations for
comparison with experiment was the program of (i) above with the
addition of a wavenumber dependent damping of the form 2.60. We
will outline the procedure used to calculate the absorption with
the assumptions of specular scattering, wavenumber dependent
permeability and no curvature of electron orbits. For a
discussion of the calculation of the absorption if diffuse
scattering is assumed the reader is referred to the paper of
Hirst and Prange[9].
The same geometry was used for this calculation as was used
for the calculation of Section 2.2, see Figures 2.1 and 2.4.
The sample is assumed to form a slab of infinite extent lying in
the x-y plane. The front surface of the slab is at z = 0. The
slab thickness is much greater than the microwave skin depth so
that the sample may be considered semi-infinite in the
z-direction. The applied field lies in the sample plane and
points in the x-direction. We consider only cases where the
applied field is parallel to a principal axis and assume that
the magnetization is parallel to the applied field for all
values of the applied field. The calculation will be valid for
all field values if the field is along (111), for fields. greater
than IKII/Ms if the field is along (110) and for fields greater
than 21K11/Ms if the field is along (100). see Section 2.3. MCA
is included through the effective fields a and y , see Table 2-1,
Microwaves travel in the +z-direction with the electric field in
the x-direction and the magnetic field in the y-direction.
As was demonstratedFin Section 2.2 the power absorbed by
the specimen is proportional to the real part of the surface
impedance. The surface impedance is the ratio of the electric
and magnetic fields at the surface of the metal. With the
geometry of Section 2.2:
In Section 2.2 this quantity was found by ( i ) solving the
- Landau-Lifshitz equation for the permeability D ; (ii) combining
this permeability with Maxwell's equations to determine the
wavevectors of the waves which could propagate in the metal; and
(iii) solving the boundary value problem of the reflection of
microwaves from the surface of the metal. With a wavenumber
-100-
dependent conductivity (ii) is not simple as in the local
conductivity case as we cannot use Ohm's law to relate the
current density in the metal to the electric field. ~nstead we
work with the Fourier transforms of the fields in the metal and
use the generalized Ohm's law (2.58).
To describe the approach taken when assuming specular
scattering we can do no better than to quote ~ippard[8]:."With
specular scattering electrons leaving the surface have suffered
an energy change exactly as if they had come straight through
the surface from an identical semi-infinite metal in which the
real electric field is mirrored. We may therefore replace the
real problem by one in which the metal is infinite and
ex(-z) = ex(z). There will be a discontinuity in the gradient
of ex at z = 0 which means that there must be a current sheet 1
supplied by an external source at z = 0 in order to produce any
field in the infinite metal." Pippard was considering the case
in which there was no applied magnetic field so that the
electron orbits were not curved. We have a magnetic field but
are neglecting the curvature of the orbits and so may us'e the
same replacement. Since ex(-z) = ex(z), from Faraday's law
h (-2) = -h (z) and there will be a discontinuity in h at Y Y Y z = 0.
Maxwell's equations are:
where the displacement current has been neglected as in
Section 2.2. The fields are assumed to vary in the z-direction
only and to have a time dependence exp(-iot) so:
If Ohm's law were valid we would have jx = ooex and we would
recover the results of Section 2.2. We take the Fourier
transform of these equations by multiplying by exp(iqz) and
integrating from z = -a to z = +a. Remembering the
discontinuity at z = 0 we have:
OD
where f (q) = exp(iqz)f(z) and ho is the value of the magnetic
field at z = O+. Combining these equations:
Since:
then:
and the surface impedance is:
The absorbed power is:
Given expressions for the wavenumber dependent conductivity and
permeability it is straightforward to carry out this integration
numerically.
The integral may be evaluated analytically in the extreme
anomalous limit, ( q l > > l ) , if the permeability is wavenumber
independent, ie if a local damping parameter is used and
exchange is neglected. In the extreme anomalous limit the
and the surf ace impedance becomest:
This expression may be combined with the permeability obtained
with the calculation of Section 2.3 to obtain the absorption in
the extreme anomalous limit with no exchange and a local
damping. The computer program written to perform the
calculation of the absorption assuming a wavenumber dependent
conductivity and damping gave results in good agreement with
those calculated using this expression for the surface impedance
in the extreme anomalous limit.
We had two other checks on the program. The results of the
program agreed with the results of the program which carried out
the calculation of Section 2.2 in the local limit, ql << 1. The
second check was to compare line positions and widths with the
calculated line positions and widths quoted by Hirst and
Prange[9] from their calculation of the absorption which assumed
a non-local conductivity and diffuse scattering of electrons at
the sample surface. Our results were in good agreement with
theirs.
' A factor of 4a/c is often included in the definition of the surface impedance, see for example Hirst and ~range[9].
3. EXPERIMENTAL DETAILS
3.1 Introduction
The FMR linewidth, AH, increases with decreasing
temperature and saturates at low temperatures if the resistivity
ratio is greater than approximately 30(Bhagat and ~irst[l]). We
wish to determine whether the linewidth, and hence the magnetic
damping, is anisotropic at low temperatures. To do this we
measure FMR with the external field parallel to each of the
three principal axes, (loo), (110) and ( 1 1 1 ) . .Ideally we would
measure the temperature dependence of AH for each of these three
axes from room temperature to 4.2 K where the linewidth
saturates. The greatest interest is attached to the 4.2 K
measurements where any dependence of AH on the direction of the
external field with respect to the crystal axes should be most
evident.
The FMR line becomes very broad on cooling: AH at a
microwave frequency of 24 GHz increases from 320 Oe at room
temperature to approximately 1600 Oe at 4.2 K. The peak
absorption also becomes weaker as the specimen is cooled and at
4.2 K is approximately 7% that at room temperature (based on the
calculations outlined in Chapter 2 ) . The conventional method
for measuring FMR uses a field modulation technique which ,"
/
measures the derivative of the absorption with respect to the
applied field. If a constant modulation amplitude is maintained
the signal at 4.2 K is smaller than the room temperature signal
by a factor greater than 50. In practice the modulation
amplitude is reduced at low temperatures by screening due to
eddy currents. Using this technique we were able to observe FMR
at the lowest temperature accessible with liquid nitrogen
(pumped liquid nitrogen = 60 K), but could not see any signal at
4.2 K. For the 4.2 K measurements the absorption of the sample
was measured directly using a bolometer. Measurements of the
linewidth were not made between 4.2 K and 60 K, however the
resonance field could be measured over the entire temperature
range by monitoring the dc voltage on the microwave diode.
In this chapter we discuss (i) the samples and their
preparation; (ii) detection of the FMR signal; (iii) the 24 GHz
microwave cavity and the sample mounting; (iv) the 24 GHz
microwave system and (v) measurements at other frequencies.
Useful references for this chapter are "Technique of
Microwave Measurements" by Montgomery[53] and "Microwave
Measurementsw by GinztonL541 for the properties of microwave
components and resonant cavities and "Electron Spin Resonance"
. by Poole[55] for information on all experimental aspects of
magnetic resonance studies. Any unreferenced statements in this
chapter may be traced to one of these three books.
3.2 Samples
The quality and preparation of a sample used in an FMR
experiment is extremely important for obtaining reliable
results. Strains, imperfections and deviations from flatness
lead to broadening of the FMR line which obscures the intrinsic
contribution to the linewidth. We cite for example the '
experience of Frait and MacFaden[26] with Nickel. Even with
careful preparation of the sample they obtained linewidths some
200 Oe larger than the intrinsic linewidth (at 25 GHz).
The samples used for the present experiments were thin
disks cut with a (110) direction normal to the plane of the
disk. The (110) plane contains the three principal axes ( 1 0 0 ) ~
(410) and ( 1 1 1 ) (see Figure 2.2). The starting material was a
boule of single crystal Nickel, 3/4 inches in diameter, nominal
purity 99.99%, purchased from Mono crystalst. The residual
resistivity ratio (RRR = P295/~4) of this material measured on
one of the samples used for FMR measurements was 38. Bhagat and
HirstCl] found that the linewidth at 4.2 K was independent of
the resistivity ratio if this ratio was greater than
approximately 30.
The boule was oriented with x-rays using the Laue back
reflection technique. The error in alignment (angle between the
sample normal and a (110) axis) was less than 1.5 degrees.
Slabs approximately 1 mm thick were spark cut from the boule,
'~ono Crystals, 1721 Sherwood Blvd., Cleveland, Ohio
then spark cut into circular disks 16 mm in diameter, the
largest diameter consistent with the microwave cavity used. The
disks were mechanically polished, on both sides, initially with
300 grit silicon carbide paper, followed by 600 grit paper, to a
thickness of approximately 500 pm. ~pproximately 75 pm was
removed from each side alternately to a thickness of
approximately 300 pm. One side was polished with 4 pm d,iamond
grit then electropolished. Electropolishing was done in a
solution of 60% H,SO,, 40% distilled water at room temperature
with a current density of approximately 1 amp/cm2[56]. The
other side of the sample was then diamond polished to within 50
pm of the final thickness and electropolished. Sample surfaces
after electropolishing were smooth and mirror-like. The final
thickness was approximately 150 pm. This was a convenient
thickness to work with. The ratio of diameter to thickness was
100 so that the demagnetizing field (equation 2.3) was small,
being approximately 30 Oe.
The samples were not annealed. Bhagat and ~ubitz[l3] found
that the FMR linewidth at 22 GHz of well annealed samples was at
most 20 Oe narrower than samples which had not been annealed.
Also annealing usually causes the resistivity ratio to decrease,
. presumably because of incorporation of impurities(~ewar[57]).
3.3 Experimental Observation of FMR
The task is to measure the power absorbed by a sample from
an incident microwave field polarized with the microwave
magnetic field perpendicular to a static magnetic field. The
simplest way to do this is to use the sample as a termination on
the end of a piece of waveguide and monitor the reflected pow.er.
The microwave circuit for such a system would consist of a
klystron, an isolator to match the klystron to the rest of the
circuit, a directional coupler to intercept a portion of the
microwaves reflected from the sample, and a diode to detect this
signal.
If the steps outlined in Section 2.2 to obtain the
equation (2.36) were repeated using boundary conditions
appropriate for a sample in a waveguide the power reflected from
the metal surface would be found to be:
where Po is the incident power, Zs is the surface impedance of
the metal, if the conductivity is described by Ohm's law
zs =(o6/2c)/-, 6 is the skin depth, 62=c2/2noo, and Zw is the
waveguide impedance, the ratio of the maximum amplitudes of e
and h in the waveguide. The waveguide used in the 24 GHz
experiments was WR42 or RG53/U waveguide (equivalent
designations). The inside dimensions of this guide are
0.420 x 0.170 inches. The cutoff wavelength, Itc, the guide
wavelength, X and the impedance for the TElo waveguide mode are 9
(for 24 GHz, X = 1.25 cm):
X = 2x(0.420 inches) = 2.13 cm C.
hg = X/I/I-(X/X~)~ = 1.54 cm
Since Zw is close to 1 we will equate it to 1 in the essentially
qualitative discussion that follows. For Nickel at room
temperature and 24 GHz the surface impedance is:
Since the surface impedance is small (see equations 2.38):
write the absorbed power in terms of the
magnetic field at the surface of the sample since it is the
magnetic field at the surface that is known in a resonant cavity
(see below):
where f is the frequency and the integral is over the tangential
components of the microwave magnetic field. The magnetic field
at the surface is just twice the incident magnetic field.
The absorbed and reflected power vary as the external field
is swept through FMR. The fraction of the incident power
absorbed when 4 = 1 and the ratio of the change in reflected
power on sweeping through FMR to the reflected power are (using
the values of ZS quoted above):
The signal is small and is superimposed on a large background.
The signal to noise ratio may be improved by using field
modulation. A small alternating magnetic field is applied
parallel to the dc magnetic field. The component of the
reflected signal at the frequency of the modulation field is
detected and amplified by a Pock-in amplifier. If the
modulation amplitude is small compared with the FMR linewidth
the resulting signal is the derivative of the absorption with
respect to the dc field.
The choice of modulation frequency is governed by a number
of factors. Since the noise contributed by the microwave
detector (a diode in our experiments) decreases as the inverse
of the frequency a higher modulation frequency usually results
in a better signal to noise ratio. There are problems with high
frequencies however which are discussed below.
The signal may be increased by placing the sample in a
resonant cavity. For our experiments the sample formed part of
the endwall of a cavity. The sensitivity of a cavity reflection
system has been discussed by Peher[58]. A resonant cavity is
the microwave anal.ogue of a resonant LCR circuit. The cavity is
characterized by a resonant frequency, f o r and a quality factor,
Qf which relates the energy stored in the cavity. Estoredf to
the energy dissipated in one cycle:
= '* E~tored /E dissipated in one cycle
Energy is dissipated in three ways: by resistive losses in the
walls of the cavity, by absorption in the sample and by
radiation through the hole used to couple microwaves into the
cavity. The unloaded Q, QU, the external Q, Qe and the loaded
Q, QL are defined:
- Qe - 2 n Estored /E lost through coupling hole
in one cycle
where Ew is the energy dissipated in the walls of the cavity in
one cycle and Es is the energy dissipated in the sample in one
cycle. The loaded Q takes into consideration all the energy
lost or dissipated. The energy stored in the cavity is:
where the integral is over the volume of the cavity and is
evaluated at a time when-the magnetic fields are at their
maximum value (the electric fields are zero at this time). The
energy absorbed in the walls of the cavity in one cycle is Ew:
where 6, is the skin depth of the walls, the permeability of the
walls is taken to be 1 and the integral is evaluated when the
fields have their maximum value. In addition to the quality
factors defined above it is convenient to define what may be
called the sample Q, Qs, and,the filling factor, q :
Qs = 27r Estored/E dissipated in one cycle
These integrals are straightforward to work out given the field
distributions for the cavity mode of interest.
If microwave power Po is incident on the cavity the power
absorbed in the cavity, and the power reflected from the
cavity, Pr, are determined by the coupling constant 8:
If equals one the cavity is said to be 'critically coupled'.
If the klystron frequency matches the resonant frequency of the
cavity at critical coupling the reflected power is zero and all
the incident power is dissipated in the sample and the cavity
walls. If is greater(1ess) than one the cavity is
under (over )coupled.
Use of a cavity has two effects, the power absorbed by the
sample may be increased, and the sensitivity, the ratio APr/p,,
is increased over that if no cavity is used.
The ratio of the energy absorbed in the sample to the total
energy absorbed in the cavity (walls plus sample) is:
A very crude estimate of this ratio is the ratio of the 'area of
the sample to the total wall area. Suppose the sample area was
1/30 the total wall area and that the coupling of the cavity was
adjusted so that 2/3 of the incident power was absorbed in the
cavity (a typical situation). The power absorbed by the sample
would be approximately 1/45 of the incident power, a large
increase over the power absorbed if the sample formed a short on
the end of the waveguide (compare (3.2) above). If we wished to
maximize the power absorbed by the sample we would use a
. critically coupled cavity, so that all the incident power was
absorbed in the cavity, and make the area of the sample as large
a fraction of the total wall area as possible. The Q of the
cavity is irrelevant in maximizing the absorbed power.
The increase in sensitivity is a more important effect for
reflection measurements. We assume for this discussion that the
ratio of the energy dissipated in the sample to that dissipated
in the walls is small. The change in absorption of the sample
on sweeping through FMR has then only a small effect on the
cavity Q. If the energy absorbed by the sample changes by an
amount AEs the change in the unloaded Q is:
AQu = ( aQ,/aES)AEs = -QuAES/(E w +ES (3.11)
This change in Q produces a change in the power reflected from
the cavity:
For a given AE, the change in reflected power is a maximum if
fl = 2fd3, the plus(minus) sign corresponding to an under(over1
coupled cavity. The sensitivity for the two couplings is the
same. The maximum is fairly broad so that it is not necessary
that f l be exactly 2243. It is clear that critical coupling
(fl = I ) must be avoided. With f i = 2fd3 the power reflected from
the cavity when Us = 0 is 1/3 of the incident power. The
change in reflected power if AES + 0 is:
The quantity AES/(Ew+Es) may be related to the unloaded Q:
so that, at optimum coupling:
where Es has been neglected with respect to Ew. The sensitivity
now involves a factor containing the unloaded Q of the cavity.
Since values of the Q are typically several thousand this may
provide a substantial increase in signal. Note that the filling
factor, q , is important in determining the sensitivity. When
comparing cavities resonating in different modes, for example a
rectangular vs a cylindrical cavity, the filling factor must be
considered. In other words the Q isn't everything. To maximize
the sensitivity in reflection measurements we would choose a
cavity with as high a combination of filling factor and Q as
possible, coupled so that 1/3 of the incident power is
reflected.
his analysis holds only if the fraction of the power
absorbed by the sample is small in which case the power
reflected from the cavity varies linearly with the change in
absorption of the sample. If the change in power absorbed by
the sample is an appreciable fraction of the total power
absorbed in the cavity the reflected power no longer varies
directly with the absorption of the sample. The sample is said
to load the cavity. If the loading is severe the PMR lineshape
will be distorted.
The resonant frequency of the cavity is shifted by the
absorption and by the reactive component of the surface
impedance of the sample. In an absorption experiment we are
interested in the changes in power reflected from the cavity due
to the change in the cavity Q not those due to change in'the
cavity frequency relative to the klystron frequency. The
klystron frequency is usually locked to the cavity resonant
frequency. The klystron frequency can be modulated by applying
a small alternating voltage on top of the dc klystron reflector
voltage. The amplitude and phase of the component of the signal
reflected from the cavity at the modulation frequency depend on
the difference between the klystron center frequency and. the
cavity frequency because at resonance there is a decrease in the
power reflected from the cavity. For small differences the
amplitude is directly proportional to the difference. Using a
lock-in amplifier this component may be detected and used to
generate a dc voltage which is fed back to the klystron
reflector so that the klystron frequency follows the cavity
frequency.
The sample forms part of the cavity endwall. Since
microwave currents flow across the junction between the cavity
and the sample good electrical contact is essential to avoid
distorting the cavity mode. Poor contact reduces the cavity Q
and changes the resonant frequency and the coupling. These
changes may depend on the external field due to the changing
surface impedance of the sample. Therefore when FMR is measured
with a poor contact the lineshape may be badly distorted.
The energy dissipated in the cavity depends on the
electrical conductivity of the walls. The conductivity
increases with decreasing temperature leading to a higher Q, a
higher resonant frequency and different coupling. Usually it is
necessary to use a tuning rod, a piece of quartz for example,
which lowers the resonant frequency when inserted into the
cavity, to ensure that the cavity resonant frequency does not
escape the frequency range of the klystron. The cavity
frequency also shifts when the cavity is evacuated, due to the
dielectric constant of air. In our experiments these shifts
were of the order of a tenth of a gigahertz.
As stated above the field modulation technique yields the
derivative of the absorption. For very broad lines, such as
those for ~ickel at low temperatures, the sensitivity of this
technique is small. The amplitude of the extrema of the
absorption derivative varies as ( A H ) - I o 5 so for a constant
modulation amplitude the signal would decrease in the same
manner. In our case it became necessary to measure the
- absorption directly. We used a bolometer to measure the
temperature of the sample. The use of a bolometer in magnetic
resonance measurements has been discussed by Schmidt and
Solomon[59] and by Cochran, Heinrich and Dewar[60]. The field
modulation and bolometric systems used in our experiments are
described in Section 3.5.
3.4 The Cavity and the Sample Holder
The Cavity
The microwave resonant cavity which was used in these
experiments is sketched in Figure 3.l(a). The cavity consisted
of two parts: the main body, which was basically a metal bucket,
and an endwall which was clamped over the open end of the
cavity. The endwall is sketched in Figure 3.l(b) and is
described below. The cavity dimensions were 14 mm deep and
16.3 mm inside diameter. The thickness of the upper end, the
end with the coupling hole, was 0.020 inches.
The cavity resonated in the TE,,, cylindrical cavity mode.
The field distributions for this mode are shown in ~igure 3.2.
The cylindrical TE,, waveguide mode is the dominant mode for a
cylindrical geometry, the cutoff wavelength being 3.413 times
the cylinder radius. This mode is similar to the rectangular
TElo waveguide mode and may be derived from the rectangular mode
by a distortion of the rectangular guide to a cylindrical shape,
The cavity was coupled to the waveguide through the upper
endwall and the cavity was oriented so that the sample was in a
horizontal plane. The angle between the sample normal and the
vertical was less than 1'.
-120-
COUPLING HOLE
TUNING ROD HOIX
- SPRINGS
SAMPLE . .
END WALL
BRASS PLUNGER
Figure 3.1 The 24 GHz microwave cavity (a) and the sample
holder (b).
Fiqure 3.2 Field configurations for the TE,,, cylindrical cavity
mode. The dashed lines represent the microwave magnetic field
and the dots and crosses represent the microwave electric field.
The cavity was designed for a high frequency modulation
system. For this purpose it was desired to use a material with
a poor electrical conductivity which would not shield the
modulation field. The material used was a 30% Ni-70% Cu alloy
which has a room temperature resistivity of approximately
40x10-~ Ocm which is essentially temperature independent. To
improve the Q the cavity was polished with diamond polish and
plated with a layer of gold several microwave skin depths thick.
Since the high frequency modulation did not work well, however,
the use of a poor conductivity material for the cavity was not
important.
The cavity coupling was adjusted to give nearly maximum
sensitivity in reflection. As pointed out above this is not a
critical adjustment. However care must be taken to avoid the
condition corresponding to critical coupling. Critical coupling
may accidentally occur if the cavity is undercoupled at room
temperature and if the Q increases upon cooling. The resonant
frequency at room temperature was near 23.95 GHz. The loaded Q
was approximately 3500. All of these quantities depended on the
quality of the contact established between the endwall and the
cavity. Before making a measurement the clamping bolts were
adjusted to obtain the maximum loaded Q. The cavity frequency
could be varied by means of a quartz tuning rod driven
vertically into the cavity by a micrometer drive mounted on the
upper flange (see Figure 3.4).
The Sample Holder
To measure FMR it is necessary to hold the sample in a
fixed orientation with respect to the applied field. The
magnetocrystalline anisotropy becomes large in Nickel at low
temperatures, the first MCA constant increasing from a room
temperature value of -0.59x105 erg/cm3 to -12.9x105 erg/c.m3 at
4.2 K. This corresponds to an effective field of approximately
2.5 kOe at 4.2 K. If the applied field is not parallel to an
easy axis (( 1 1 1 ) direction) large torques arise which tend to
rotate the sample until an easy axis is parallel to the field.
If the sample is prevented from rotating by being glued or
soldered to a substrate the strain resulting from the
differential thermal contraction of the sample and the substrate
leads, through magnetostriction, to shifts in the position of
FMR and may lead to broadening of the line. The mounting of the
sample is thus of critical importance for low temperature
measurements.
Our solution to this problem is shown in Figure 3.l(b).
The sample was held in a demduntable endwall assembly which was
clamped over the open end of the cavity. The endwall was a
- circular piece of copper 1/4 inch thick. The surface forming
part of the cavity was gold plated. The center of the endwall . was machined to 8.020 inches thick to accommodate the sample.
The center of the sample was exposed to the microwaves through a
hole 7 mm in diameter. The sample was lightly pushed against
this wall by a brass plunger attached to the endwall. Springs
were placed on the bolts holding the plunger to the endwall to
avoid having the pressure on the sample vary with temperature
due to the thermal contraction of the various pieces. These
springs were wound from phosphor bronze wire. The plunger had a
hole in it to allow access to the back of the sample for the
measurements made with the bolometer (Section 3.5).
The endwall was clamped to the cavity using the two rings
shown in the figure. The bolts used here were also spring
loaded. The endwall assembly could be rotated on the cavity;
thus rotating the sample in order to measure FMR along each of
the three principal axes was straightforward and meant that the
sample mounting remained unchanged between coolings to 4.2 K.
The endwall also fitted a cavity which was part of a 9.5 GHz
system so that measurements could be made at the two frequencies
without having to remount the sample.
3.5 The 24 GHz Microwave System
A schematic drawing of the 24 GHz system is shown in Figure
3 , 3 . The part of the system to the left of the vertical dashed
line in the Figure was part of a microwave transmission system
which has been described in detail[60], with the addition of the
electronic switch. Apart from the klystron and the klystron
power supply all the microwave components were the same as those
described in that paper. The reader is referred to that paper
Fiqure 3.3 Schematic drawing of the 24 GHz microwave system.
if more information is required.
The components were mounted on a table approximately 2 m
high from which the resonant cavity was suspended in the magnet
gap. The magnet was mounted on rails and could be moved to
allow easy access to the cavity. For low temperature
measurements a stainless steel liquid helium dewar was placed
around the cavity and bolted to a flange on the table. .
The magnet was a Varian V-3800 electromagnet having a
3 1/2 inch gap. Fields up to 16 kOe could be obtained. A
Bell 810 Field Meter was used to provide a signal proportional
to the value of the field. The field values were calibrated
with an NMR system[61]. In addition to being mounted on rails
the magnet yoke could be rotated about a vertical axis.
Microwaves were generated by a Varian klystron (type
VA 282 EY) driven by a PRD Electronics Inc Type 819-A Universal
Klystron Power Supply. The klystron operated in a frequency
range 23.8' to 24.0 GHz with an output power of 300 mW. The
klystron frequency was locked to the resonant frequency of the
microwave cavity as described above.
The electronic switch was used to amplitude modulate the
microwave power for measurements made with the bolometer (see
below). The four port switch was not essential for the FMR
measurements but was useful for diverting the microwaves when
changing the sample. The signal reflected from the cavity was
detected by the microwave diode attached to the directional
coupler. The diode mount was electrically isolated from the
microwave track by placing a mica gasket between the waveguide
flanges and using nylon bolts. This overcame problems with
ground loops. The microwave frequency could be measured with an
accuracy of .005 GHz by means of a Hewlett Packard K532A
frequency meter. The variable attenuator served to vary the
power incident on the cavity.
The part of the system to the right of the dashed line in
Figure 3.3 is shown in Figure 3.4. This part of the system was
designed specifically for these experiments. A vacuum seal
consisting of a mica gasket and a Viton O-ring was placed
between the microwave flanges above the upper flange. A length
of stainless steel waveguide between the upper flange and the
cavity provided for thermal isolation of the cavity. The
copper-stainless steel waveguide joint was made by milling out
the inside of a portion of the copper waveguide and soldering
the stainless guide to it.
The cavity was bolted to a flat flange at the end of the
guide, and could be easily removed for changing samples and
,mounting the bolometer. A stainless steel can, 2 inches in
diameter, could be attached to the lower flange to isolate the
cavity for low temperature measurements. This can could be
. evacuated through the stainless steel tube shown in the figure.
The tuning rod for the cavity and wires were brought into the
can through this tube. Two additional vacuum feedthroughs were
placed in the lower flange. A heater resistor was attached to
the waveguide above the cavity to allow one to vary the
TO KICROKETBR D R I VE /FOR TH'I TUNIIVG ROOD
VACUUM S U L \ - /
FEEDTHROUGHS
FOR HELIUM TRANSFER TUBE
HEATER' R E S I S T O R -1
VACUUM PUMP
-UPPER FLANGE
-TO VACUUN PUMP AND HELIUM RECO VERY L I NZ
COPPER WAVZGUIDE - S T A I N L E S S S T E E L WAVM;UIDS
- - S T A I N L E S S S T E E L T U B E FOR EVACUATING S P A C E AROUND T H E C A V I T Y
- R A D I A T I O N S H I E L D S
/ LOWER FLANGE FOR ATTACHING CAN
/ TUNING ROD D R I V E
, CAVITY
Fiqure 3.4 P a r t o f t h e 24 GHz microwave sy s t em.
temperature of the cavity.
Temperatures were measured using a copper constantan
thermocouple and using a carbon glass resistance thermometer
attached to the cavity endwall. The resistance of the carbon
glass thermometer was measured by means of a Keithley digital
multimeter in four-wire mode or with an SHE Conductance Bridge.
A calibration table was supplied by the manufacturer.
Measurements between 4.2 K and approximately 60 K were made
on the fly as the system warmed from 4.2 K, At 60 K
measurements were made by pumping on liquid nitrogen around the
can. Temperatures above 7 7 K could be held constant by means of
a controller which regulated the current through the heater
resistor in order to maintain a constant signal from the
thermocouple. The controller kept the temperature constant to
better than 0.5 K. Since both the magnetocrystalline
anisotropy, which shifts the position of the resonance, and the
damping are strong functions of temperature between 4.2 K and
room teGperature it is essential that the temperature be held
constant. For example the effective magnetocrystalline
anisotropy field, 2 1 ~ 1 I/M~, changes by approximately 12 Oe per
degree at 200 K and by approximately 24 Oe per degree at 100 K.
The linewidth changes by approximately 12 Oe per degree at 7 7 K.
For temperatures accesible using liquid nitrogen, down to
approximately 60 K by pumping on the liquid, low frequency
(=I00 Hz) field modulation was used. No signal could be
observed at 4.2 K using this technique. An attempt was made to
use high frequency modulation but this was unsuccessful (see
below). For the 4.2 K measurements we used a bolometer to
detect the absorption directly.
The dc voltage across the microwave diode was monitored in
all measurements made. FMR was observed as a change in the dc
level. The variation with the external field could be traced on
an X-Y recorder and was used to determine whether the sample was
loading the cavity. The signal could be used to find the
resonance field Hfmr, but it was too noisy to yield reliable
values of the FMR linewidth.
Low Frequency Modulation
A pair of Helmholtz coils, approximately 30 cm in diameter,
were mounted on the pole pieces of the magnet. These were
driven by the reference channel of a PAR 124 lock-in amplifier,
amplified by a Kepco Bipolar Operational Power Supply/~mplifier.
These coils produced a field of approximately 1 Oe per volt of
driving at 100 Hz in an empty gap. The Kepco power supply could
deliver 75 volts. The field amplitude at the sample was less
than 1 0e/volt because of screening of the field by eddy
currents in the cavity walls and in the dewar. The dewar
contained a liquid nitrogen cooled copper shield around the
helium pot so the screening became appreciable at low
temperatures, even at a frequency of 100 Hz.
The voltage from the microwave diode was fed to the lock-in
amplifier. The output from the lock-in amplifier went to the
Y-channel of an X-Y recorder. The input to the X-channel was
obtained from a Hall probe which provided a voltage proportional
to the external field. Since a field modulation technique
yields the derivative of the absorption with respect to the
external field the linewidth, defined as the field interval
between extrema of the derivative, and the resonance field, the
zero crossing of the derivative, could be read directly from the
X-Y recorder trace. Traces were taken at least twice, sweeping
in the direction of both increasing and decreasing values of the
external field, in order to check reproducibility and in order
to check that the field sweep rate was sufficiently slow
compared with the lock-in amplifier time constant so that the
absorption line was not distorted by too fast a sweep rate.
High Frequency Modulation
The signal to noise ratio in a field modulation system may
be improved by using a higher modulation frequency as the noise
contributed by the detector varies approximately inversely with
the frequency. The use of high frequency modulation is attended
with problems. The modulation field must penetrate to the
inside of the cavity, however the penetration decreases with
increasing frequency, The current must be increased,or the
modulation coils placed close to the sample, to achieve the same
modulation amplitude at a high frequency as at low frequencies.
Since the skin depth decreases with increasing conductivity the
problem becomes more severe at low temperatures. Eddy currents
are induced in the cavity walls and in the sample by the
modulation field. The interaction between these eddy currents
and the external field causes the walls and the sample to
vibrate. Essentially, the resonant frequency of the cavi'ty is
modulated. This gives rise to a signal in the reflected
microwaves at the modulation frequency proportional to the
strength of the external field. This field dependent background
may become large and obscure the FMR signal. Heating due to the
eddy currents may result in the temperature of the sample
drifting with time or it may result in excessive boil-off of
liquid Helium. Due to these problems we were unable to
construct a high frequency modulation system which worked as
well as the low frequency modulation system.
The Bolometer
A bolometer is a chunk of material whose electrical
resistivity depends in some known way on its temperature. The
temperature of the sample increases slightly with the power
absorbed. The change in temperature can be detected by
measuring the resistance of a bolometer attached to the sample.
The bolometer which was used was purchased from Infrared
Laboratories lnci. It is sketched in Figure 3.5. A piece of
Germanium approximately 0.4 mm square was attached to Indium
blobs on a sapphire substrate. Brass leads were attached to the
Indium in order to measure the bolometer resistance. In
operation the sapphire substrate was attached to the
ferromagnetic sample (see below). The bolometer resistance was
15 52 at room temperature, approximately 400 51 at 12 K, 21.0 KQ-at
4.4 K and 250 KQ at 4.2 K. The change in resistance on a sweep
through FMR was approximately 1 KQ, small enough so that the
dependence of the resistance on temperature was essentially
linear.
The bolometer was placed in series with a 9 volt battery
and a 1.5 MQ resistor. The microwave power was chopped with the
electronic switch (see Figure 3.3). A chopping frequency of
80 Hz worked well. The chopper was driven by the reference
channel of a PAR 122 lock-in amplifier and the voltage across
the bolometer provided the input to the lock-in amplifier. The
signal was observed at the chopping frequency. The amplifier
output went to the Y-channel of an X-Y recorder and to a
data-acquisition system where the data was stored in digital
form. The data could be transferred to the main SFU computer
for analysis (see Chapter 4 ) . The program used in the data
acquisition required a zero level for scaling the data. This
was provided by shorting the input to the lock-in amplifier
'~nfrared Laboratories Inc, 1808 E 17th St, Tucson ~rizona, 85719
INDIUM BLOBS
BRASS LEADS F'HIRX SUBSTRATE
1/16" DIA x 0.004"
Fiqure 3.5 The bolometer which was used for the 4.2 K
measurements, approximate scale 1 inch = 1 mm.
(equivalent to turning off the microwave power). The program
then scaled the data so that the maximum signal was 1.
The power reflected from the cavity was monitored using the
microwave diode to determine the variation of the power
reflected from the cavity. The reflected power, and hence the
power in the cavity, changed by less than 0.7% during a field
sweep. Since this power variation was so small the differential
technique of Cochran et al[60] using two bolometers was not
needed.
The procedure for making a measurement was as follows. The
sample was placed in the endwall assembly, the endwall attached
to the cavity and the sample orientation determined by measuring
the angular variation of the resonance field. When the desired
crystal axis was located the endwall was rotated until that axis
was parallel to the applied field when the applied and microwave
magnetic fields were perpendicular. The cavity was removed from
the waveguide and the bolometer attached to the back of the
sample with dilute GE 7031 adhesive. The cavity was reattached
to the waveguide, the stainless steel can flushed with Helium
gas and attached to the the lower flange and precooling started.
The can was not evacuated until the temperature fell to
approximately 120 K in order to avoid drying out the adhesive.
It was desirable that the adhesive remain semi-fluid during
cooling so as not to strain the specimen. The Helium transfer
was then started.
The pressure of Helium gas in the can while using the
bolometer was critical for reproducible results. It appears
that the thermal contact with the liquid Helium bath provided by
the gas is an important effect. Best results were obtained with
a pressure of approximately 1 torr. The system usually worked
well although there were occasional drifts with time and sudden
jumps in the signal level that remain unexplained. A suf'ficient
length of time spent fiddling with the pressure in the can
usually cured these problems. A second reference bolometer
would have been of great help in order to extract the sample
signal from these background noises.
Several field sweeps were made, in the direction of both
increasing and decreasing field, to check reproducibility, When
the system was working properly the reproducibility was good.
In the early experiments with the bolometer the bolometer
was left attached to the sample(samp1e 1 ) for four successive
coolings to 4.2 K. Upon removal of the sample from the endwall
assembly a small dimple was noted where the bolometer had been
attached. For all subsequent measurements the bolometer was
removed and reattached between coolings. Measurements on the
dimpled sample were in agreement with those on an undimpled
sample(samp1e 2 ) . For sample 2 measurements were made at room
temperature and at 7 7 K before attaching the bolometer. The
4.2 K measurements were then made and the room temperature and
77 K measurements repeated. The results before and after
cooling to 4.2 K were the same. We conclude that the attachment
of the bolometer to the sample did not produce any strain in the
sample which would have led to shifts or broadening of the FMR
line.
3.6 Measurements at other Frequencies
Measurements were made at 9.1, 34.7 and 73.0 GHz at room
temperature and at 9.5 GHz at 4.2 K in addition to the
measurements at 24 GHz, The 9.5 GHz system was identical to the
24 GHz system except that a circulator was used to separate the
reflected microwaves from the incident microwave power instead
of a directional coupler. The cavity was of the same
construction as the 24 GHz cavity, although of a different size
of course, and resonated in the same mode (TE,,,). As mentioned
in Section 3.3 the sample mount fitted both the 9.5 and 24 GHz
cavities so that measurements could be made at both frequencies
without having to remount the sample.
Room temperature measurements at frequencies other than
24 GHz were made without resonant cavities. The sample formed
part of a termination at the end of a piece of waveguide. A
circular area 3 mm in diameter was exposed through a 0.005 inch
thick copper diaphragm for the 9.1 and 34.7 GHz measurements.
The sample was placed directly across the waveguide
( 2 mm x 3.5 mm) at 73 GHz. Field modulation at a frequency near
20 KHz was provided by a wire passing directly underneath the
sample.
4 . EXPERIMENTAL RESULTS AND DISCUSSION
4.1 Introduction
In this chapter we present and discuss our experimental
results. The measurements were made primarily to investigate
the magnetic damping in Nickel at low temperatures. There are
two main thrusts to our work. First, we wish to determine
whether or not the FMR linewidth, AH, is different with the
applied field parallel to each of the three principal crystal
axes at low temperatures. Second, we are interested in the
information about the damping processes that can be obtained by
comparing the experimental FMR line widths, positions and shapes
with widths, positions and shapes calculated using computer
programs based on the calculations outlined in Chapter 2.
We have made measurements on two samples cut from the same
boule of Nickel and prepared in the same way. FMR was measured
with the applied field in the sample plane and along each of the
three principal axes at 23.95 GHz at room temperature and from
4.2 to 200 K . As stated in Chapter 3 we were unable to obtain
values for the FMR linewidth between 4.2 and approximately 60 K.
The variation of the resonance field, Hfmr, with the direction
of the applied field in the sample plane, for directions in
addition to the three principal axes, was measured at room
temperature, 77 K and 4.2 K . The angular variation ~f the
linewidth was measured at 77 K. Also, the frequency dependence
of FMR was measured at room temperature for Sample 2. A
measurement was made at 9.5 GHz at 4.2 K, but no other
measurements were made at other frequencies at temperatures
other than room temperature. Low temperature systems were not
available for frequencies other than 9.5 and 23.95 GHz. At
9.5 GHz the magnetocrystalline anisotropy shifts mean that FMR
can be observed only for the ( 1 0 0 ) direction. The results are
presented in the following Sections: room temperature results in
Section 4.2; 77 K results in Section 4.3; 4.2 K results in
Section 4.4; and in Section 4.5 the measurements made at
intermediate temperatures. We discuss the results briefly in
each Section but postpone a more comprehensive discussion until
Section 4.6.
Before presenting the results it is worthwhile to summarize
the calculations available for comparison with experiment. If
damping and exchange are neglected FMR occurs at the field where
the permeability becomes infinite. From Chapter 2 this is when:
where a and y are effective MCA fields, see Table 2-1 and
equations ( 2 . 5 7 ) . Ho is the applied field and Hd is the
demagnetizing field. The value of the applied field which
satisfies this relation will be referred to as the 'no-exchange
no-damping' value of the resonance field. The experimental
value of the resonance field will differ from this value of
course, because of the shifts due to damping and exchange. We
will be interested in comparing the experimental resonance
fields with calculated resonance fields and in such a comparison
this no-exchange no-damping value forms a useful reference
point.
Three computer programs incorporating different options
were used for calculating the absorption and the absorption
derivative. The first program, to be referred to as program I ,
used a local conductivity, exchange, wavenumber independent
Gilbert damping and magnetocrystalline anisotropy (MCA) . In
this program it was assumed that the applied field was parallel
to a crystal axis and that the magnetization was parallel to the
applied field. This calculation was outlined in Section 2.2.
The second program, which will be referred to as program 11,
used a local conductivity, Gilbert damping and neglected
exchange. The absorption was calculated for arbitrary
orientation of the applied field with respect to the sample
plane and the crystal axes and allowed for the lack of alignment
between the magnetization and the applied field. Exchange was
neglected because the calculation becomes quite complicated and
exchange is a comparatively small effect in Nickel. This
program was used primarily to determine the effect on the
resonance line of misalignment between the applied field and a
crystal axis. It was also useful for calculating the variation
of the resonance field with the direction of the applied field
in the sample plane. This calculation was outlined in
Section 2.3. The third program, which will be referred to as
program 111, assumed, as in program I, that the applied field
was parallel to a crystal axis and that the magnetization was
parallel to the applied field. Exchange and MCA were included.
The program incorporated the option of a local or a non-local
conductivity and a wavenumber dependent or independent Gilbert
damping. This calculation was outlined in Section 2.4. The
program used a Fourier sum to evaluate the absorption and
absorption derivative. The results of this piogram agreed with
those of programs I and I1 in the limits where they could be
compared of course. This is the program which was used for most
of the discussion which follows.
Material Parameters for Nickel
A large number of factors enter these calculations which
serve to determine the experimentally observed quantities.
These include the saturation magnetization, the g-factor, the
exchange constant, the.dc conductivity and its dependence on
wavenumber, the damping parameter and its dependence on
wavenumber, and the magnetocrystalline anisotropy constants. In
addition the microwave frequency, the sample size and shape, the
direction of the applied field with respect to the crystal axes,
and the temperature all have profound effects. In principle it
is possible to obtain values for many of the material parameters
entering FMR from the FMR measurements. Our interest is
primarily in the damping and so we take values for most of the
parameters from the literature. It is convenient to collect the
parameters which we will use in our discussion in one place.
The parameters used in the calculations at room temperature, 77
and 4.2 K are listed in Table 4-1. The values of the saturation
magnetization were taken from the work of Kaul and ~hompson[62]
and Danan, Herr and ~eyer[63]. The resistivity ratio at 4.2 K
was measured on Sample 1 , and was found to be p2,,/p4=38. The
resistivity at any temperature was assumed to be given by the
sum of a constant residual resistivity and the resistivity that
would be observed in an ideally pure Nickel sample[64,65,66].
The values of the condu=tion electron relaxation time 7 enter
the calculations made using a non-local conductivity. Following
Cochran and Heinrich[37] we have assumed a room temperature
value of lo-" sec and a value for the Fermi velocity vF
= 2.5x107 cm/sec. These correspond to a room temperature mean
free path, I = vFr, of 25 A. These values were extracted from
the low temperature cyclotron resonance data for s-p band belly
orbits as reported by Goy and Grimes[68]. This relaxation time
varied with temperature in the same way as the dc electrical
conductivity. The values of the damping parameter are listed
for comparison purposes as we will vary the damping parameter
when comparing calculations with experiment. The room
temperature value is that of Dewar, Heinrich and cochran[381
while the low temperature values are taken from the work of
TABLE 4-1
Nickel Parameters
295 K 77 K 4.2 K
4rm~~(kG) 162,631 6.16 6.60 6.60
B = 7.4x105 erg/cm3; @, = 17": SB@~/M~ = 180 Oe
K1' = Kt + 5~@2/2 = -12.44X105 erg/cm3[21]
Temperature Independent Parameters:
g = 2.187[38]
w / y = 7.82 kOe at f = 23.95 GHz
A = 1 . 0 ~ 1 0 - ~ erg/cm[38,13]
v = 2.5x107 cm/sec[37] F
Demagnetizing field Hd = 30 Oe
Bhagat and ~ubitzE131. The numbers were taken from Figure 16 of
[13](an enlarged version of Figure 6 of [12]). These are
actually values of the Landau-Lifshitz damping parameter, but,
as pointed out in Chapter 2, the difference between the
Landau-Lifshitz and Gilbert damping parameters is small. ,
The value of the g-factor is due to Dewar, Heinrich and
Cochran[38] and was derived from FMAR transmission measurements,
The values of g quoted by different authors[24,38] are
independent of temperature. The exchange constant is that used
by Cochran, Heinrich and ~ewar[48] and Bhagat and ~ubitz[l31.
The Fermi velocity was discussed above. The demagnetizing field
was calculated using the formula of Kraus and ~rait[42], .
equation (2.5). A temperature independent value of 30 Oe was
used. The demagnetizing field shifts the resonance but has no
effect on the FMR lineshape if the applied field is in the
sample plane.
The MCA constants listed in the Table are those of
~okunaga[67] at room temperature and Tung, Said and ~verett[21]
at 77 and 4.2 K. The room temperature MCA constants of Tokunaga
are in good agreement with those of ~ranse[28]. In the past
there has been wild disagreement about the values of the MCA
constants, particularly the higher order constants K2 and K3 at
low temperatures[28]. However some accord seems to have been
reached. The constants of Tung et a1 at 4.2 K are in good
agreement with those obtained by ~ersdorf[44] from the torque
measurements of Aubert et a1[43]. The constants of Tung et a1
at 77 K are in good agreement with those of Franse[28]. However
Tokunaga's constants at these two temperatures do not agree with
these values. For example at 77 K Tokunaga has
K2 = -1.4x105 erg/cm3 and K3 = .28x105 erg/cm3 which are of the
same order of magnitude but have opposite signs to the constants
of Tung.et a1 and of Franse. At 4.2 K Tokunaga has
K2 = 2.0x105 erg/cm3 and K3 = 3.3x105 erg/cm3 which have the
same sign as the constants of Tung et al. These differences are
important because the calculated values of the resonance field
will be shifted depending on which MCA constants are used.
At intermediate temperatures the only data available is
that of ~okunaga[67]. Since these constants do not agree 'with
those of Tung et a1 at 77 K we will be careful in the
conclusions we draw from their use.
At 4.2 K there is an additional contribution to the MCA
which has been ascribed to the presence of a small piece of
Fermi surface, the X,'hole pocket, which exists only when the
magnetization is within an angle qjO of a (100) direction.
Gersdorfi441 has suggested that the extra free energy
contributed by this piece of Fermi surface is:
if the angle, betweeh the magnetization and a (100) direction,
4, is less than 4, and zero otherwise. This additional energy
produces a torque which enters the effective MCA fields a and 7.
If the magnetization is parallel to a (100) direction the
effective MCA fields are (compare Table 2-11:
where K.1' = K 1 + 5B4;/2. This result is obtained by taking the
second derivative a2E/ad2 evaluated at 4 = 0 . The effective MCA
fields for the ( 1 1 1 ) and ( 1 1 0 ) directions are not changed since
the X2 pocket does not exist if the magnetization points along a
( 1 1 1 ) or a ( 1 1 0 ) direction. The values of K 1 ' and 5 ~ 4 : / ~ ~ are
listed in the Table using the values of Tung et a1 for B and do.
These are in good agreement with Gersdorf's values for these
parameters. The effect of this extra MCA torque is to shift the
position of the ( 1 0 0 ) resonance some 200 Oe to lower fields.
~ l t h o u g h the suggested presence of the X, pocket has resolved
problems with the description of MCA in Nickel at 4 . 2 ' ~ , the
actual existence of the pocket is still not a certainty. For
example it has not been observed in de Haas van Alphen
experiments[69]. It should be pointed out that the MCA torques
for the three principal axes at 4.2 K may be calculated directly
from the Fourier coefficients of the torque curves given by
Aubert et aP[43]. The values of a and y obtained in this manner
agree closely with those obtained using the MCA constants of
Tung et a1 and of Gersdorf, as they should since Gersdorf used
Aubert's numbers to obtain his MCA constants.
Gersdorf and Tung et a1 quote values for K4 at 4.2 K which
do not agree at all. Since K4 has only a small effect on the
position of FMR we have neglected it in our calculations.
4.2 Room Temperature Results
For measurements at temperatures above 60 K FMR was
measured using a field modulation technique, see Chapter 3. The
result of an experiment was an X-Y recorder trace of a signal
proportional to the derivative of the power absorbed by the
sample as a function of the applied field. The linewidth, AH,
was measured directly from the recorder trace as the field
interval between the extrema of the derivative. The resonance
field, Hfmr, was measured as the zero crossing of the
derivative. FMR was also measured by monitoring the dc voltage
across the microwave diode which provided a signal proportional
to the absorbed power. -The resonance field could be obtained
from the maximum of this absorption signal.
The experimental values of the resonance field and the
linewidth for the two samples at room temperature and at
23.95 GHz are listed in Table 4-2, along with the no-exchange
no-damping values of Hfmr. The results for the two samples are
generally in good agreement although the ( 1 1 0 ) linewidth for
sample 2 is larger than the other measured linewidths.
The Frequency Dependence of FMR
The linewidth calculated using program 1 with
G = 2.45x108 sec-I at 23.95 GMz is 320 Oe. This is the
linewidth expected for a sample in which the linewidth was due
TABLE 4-2
Results for room temperature, 23.95 GHz.
[loo1 [1101 6 1 1 1 1
Hfmr ( kOe) Sample 1 5.58+0.02 5.35 5.14
Sample 2 5.58 5.34 5.14
Calc. 5.60 5.38 5.17
AH(0e) Sample 1 350230 365 360
Sample 2 340 380 360
Calc: No-exchange, no-damping value of Hfmr.
In this and all subsequent Tables the quoted experimental
uncertainties apply to all entries in the ,Table.
only to the intrinsic damping and the exchange conductivity with
no surface anisotropy. The average linewidth we have measured
at this frequency is 360 Oe which is 40 Oe larger than the
'ideal linewidth' of 320 Oe. The linewidth measured in an FMR
experiment may be increased over the ideal linewidth because of,
for example, strain in the sample, inhomogeneities, impurities,
surface roughness or polycrystallinity. The mechanism may be
spin-pinning, described by a surface anisotropy energy,
two-magnon scattering, or the sample may see an inhomogeneous
applied field because of, say, surface irregularities.
Measurement of the frequency dependence of the linewidth
provides a way of sorting out some of these non-intrinsic
contributions to the linewidth. We have measured FMR at
frequencies of 9.115, 34.7 and 73 GHz in addition to 23.95 GHz,
at room, temperature. These measurements were made only on
sample 2 because sample 1 was damaged slightly after the
measurement of the resistivity ratio. The measurements at
73 GHz were difficult because the signal was very small. As a
result we were able to measure only the resonance for the (100)
direction at this frequency. The resonance fields and
linewidths obtained from experiment and those calculated using
program I are listed in Table 4-3, and the frequency dependence
of the linewidth is shown in Figure 4.1. The agreement between
the experimental and calculated values of Hfmr is generally
good. The experimental linewidths are larger than the
calculated linewidths at 9.115 and 23.95 GHz. It is somewhat
disturbing that the (110) linewidth is consistently larger than
the (100) and ( 1 1 1 ) linewidths. The differences are roughly
equal to the experimental uncertainty at each frequency. It was
demonstrated by Anderson, Bhagat and ~heng[30] that the in-plane
linewidth in Nickel at 22 GHz at room temperature was isotropic,
within their experimental uncertainty of 210 Oe.
From calculations carried out with program I the frequency
dependence of the linewidth at room temperature and in the
frequency range of interest, 2 9 GHz,is:
where f is the frequency and G is the Gilbert damping parameter,
This expression has been verified for values of G between 1 and
TABLE 4-3
Frequency dependence of Hfmr and AH at room temperature,
Sample 2.
F(GHZ) Hfmr AH(0e)
( kOe
[loo] [1101 [ 1 1 1 1 [loo] [1101 [ 1 1 1 1
9.115
Expt. 1.485.02 1.28 1.07 160220 190 160
Calc. 1.46 1.31 1.03 140 140 140
Expt. 5.58f.02 5.34 5.14 340f30 380 360
Calc. 5.58 5.37 5.16 320 320 320
34.7
Expt. 8.92k.03 8.69 8.57 420f40 475 430
Calc. 8.92 8.69 8.50 450 450 450
73.0
Expt. 21.1f.l - - 900k100 - - Calc. 21.2 21 .O 20.8 920 920 920
Calc: Program I, local conductivity, exchange, local damping,
G = 2.45x108 sec-', no surface anisotropy.
4x108 get-l. The zero frequency intercept is a result of the
exchange conductivity broadening.
The frequency dependence of the (100) linewidth is shown in
Figure 4.1. The (100) data is shown because we have values for
the linewidth at four frequencies. The solid lines on the
figure are a least squares fit of this data to a straight
line (a), and the frequency dependence expected using program I
with G = 2.45x108 sec-l(b). The slopes and intercepts of these
two lines are listed in the figure caption. Comparing the
experimental slope with (4.2) we see that our data are
consistent with a Gilbert damping parameter .
G = 2.3+0.3x108 sec-l. The uncertainty in this value is large
but it agrees well with the values of other authors[38,24,27,17]
(see the numbers quoted in section 1.2). The zero frequency
intercept is 50 Oe which is some 25 Oe larger than the 25 Oe
expected from the calculations made with program I from which
equation (4.2) was obtained. A possible explanation for this
difference would be the presence of some surface spin-pinning.
If a surface anisotropy Ks = -0.1 erg/cm2 with the anisotropy
axis parallel to the static magnetization was assumed, the
frequency dependence of the linewidth would be linear with the
same slope as ( 4 . 2 ) but having a zero frequency intercept of
50 Oe. This surface anisotropy would neatly explain our (100)
linewidth data,
In additi~n to broadening the FMR line spin pinning
produces a shift in the position of the resonance. A surface
LINEWIDTH
(Qe
0 20 40 6 0
FREQUENCY (GHz )
Fiqure 4.1 Frequency dependence of the FMR linewidth at room
temperature, sample 2. The applied field was parallel to the
[ 0 0 1 ] axis. (a) Linear fit to the data, slope = 11.6 0e/GHz,
intercept = 50 O e . (b) Calculated dependence using program I,
with the parameters of Table 4-1 and no spin-pinning,
slope = 12.3 o ~ / G H ~ , intercept = 25 Oe.
anisotropy of this magnitude would shift FMR to lower fields by
approximately 25 Oe at 9 GHz, approximately 30 Oe at 23.95 GHz,
and by approximately 35 Oe at 34.7 GHz. These shifts are
roughly equal to the uncertainty in the experimental values of
Hfmr . We note that Bhagat and Lubitz[l31 used a surface anisotropy of 0.1 erg/cm2 in their analysis.
Comparison of Experimental and Calculated Lineshapes
A typical FMR derivative curve at 23.95 GHz is shown in
Figure 4.2. This curve was measured on sample 2 with the
applied field parallel to the (100) axis. The solid line on the
figure is the absorption derivative calculated using program I ,
with a Gilbert damping of 2.6x108 sec-' and no surface
anisotropy. This value of the damping parameter was chosen to
reproduce the linewidth of 340 Oe, The other parameters used in
the calculation are those listed in Table 4-1. If we assumed a
Gilbert damping of G = 2.45x108 sec-I and a surface anisotropy
Ks = -0.1 erg/cm2 the calculated linewidth would equal the
experimental linewidth and the match between the lineshapes
would be comparable to that shown in the figure. The peaks in
the experimental curve near zero field are associated with
domain wall motion during saturation of the sample. The low
field zero crossing of the derivative occurs at Ferromagnetic
Antiresonance (FMAR). The asymmetry of the experimental
derivative, the ratio of the low field derivative peak amplitude
A B S O R P T I O N
D E R I V A T I V E
(ARB. UNITS )
-- -
o 2 4 6
, - APPLIED F I E L D (kO e )
Fiqure 4.2 Experimental FMR absorption derivative at room
temperature, 23.95 GHz, with the applied field parallel to
[OOl]. The solid line is the absorption derivative calculated
using program I, with a Gilbert damping parameter
G = 2.6x108 sec-'. The other parameters used in the calculation
are listed in Table 4-1.
to the high field derivative peak amplitude, does not agree with
the calculated asymmetry. This is a common observation. For
example Bhagat, Hirst and Anderson[27] quote experimental
asymmetries in Nickel of 1.32 to 1.40 at 22 GHz which may be
compared with their calculated asymmetry of 1.18. Our
asymmetries are similar in magnitude to those of Bhagat, Hirst
and Anderson.
The Angular Variation of
The variation of the resonance field with the direction of
the applied field in the sample plane is shown in Figure 4.3.
The data was collected by measuring FMR with the magnet rotated
from a position where the applied field was parallel to a
principal axis and perpendicular to the microwave magnetic
field. The signal becomes small, and the lineshape may be
distorted, if the angle between the applied and microwave
magnetic fields becomes small so data may be taken only in a
limited range of angles about each principal axis. The solid
line in the figure is the no-exchange no-damping value of the
resonance field calculated using the MCA fields a ( $ ) and
see equations (2.57). The agreement between the calculation and
experiment is splendid. This plot is useful primarily because
it indicates that the sample plane does indeed coincide closely
with a ( 1 1 0 ) normal crystal plane.
ANGLE (DEGREES)
Fiqure 4.3 Variation of the resonance field, Hfmr, with the
direction of the applied field in the sample plane, room
temperature, 23.95 GHz. The data was taken by rotating the
magnet about: ( 100 ) W ; (111) + ; (110) 0 . The experimental uncertainty is indicated by the single error bar at -50'. The
solid line is the no-exchange no-damping value of Hfmr.
As was pointed out in Section 2.3 the angular variation of
Hfmr should follow the no-exchange no-damping variation if
21K11/MS << Hfmr, that is, if the magnetization is parallel to
the applied field at FMR. This is certainly true here where
2IK11/MS = 240 Oe. Of course the experimental values of Hfmr
will not equal the calculated values because the damping and
exchange shifts are not considered in the calculation. The
angular variation of Hfmr obtained using equations (2.57) is the
same as that which would be obtained from calculations using
program 11, the only program which could be used for calculating
FMR when the applied field was not parallel to a principal axis.
Program I 1 did not include exchange so that the values of Hfm,
calculated with that program would not equal the experimental
values because of the neglect of the exchange shift.
The damping and exchange shift, the difference between the
experimental and the no-exchange no-damping values of Hfmr, is
approximately 30 Oe to lower fields in Nickel at room
temperature at 23.95 GHz, from the calculations listed in
Table 4-2. This is roughly the experimental uncertainty in
Hfmr . It is not reasonable to compare the resonance field values on the scale of Oersteds as the parameters entering a
calculation are not known with this precision. For example a
change in Kt from -.59x105 esg/em3 to -.57x105 erg/cm3,
corresponding to a change in the temperature of the sample of
approximately 2'[67], produces a shift in Hfmr of approximately
10 Oe.
4.3 Results at 77 K
The experimental values of the resonance field and the
linewidth for the two samples at 77 K at 23.95 GHz are listed in
Table 4-4. The results for the two samples were in good
agreement with each other. We found that the linewidth at 77 K
was anisotropic. The linewidths measured with the applied field
parallel to the (100) and ( 1 1 1 ) axes were identical within
experimental uncertainty. The (110) linewidth was approximately
160 Oe, or 16% larger, for both samples.
Also listed in Table 4-4 are the no-exchange, no-damping
values of HfmrI and the values of Hfmr calculated using a local
conductivity and wavenumber independent Gilbert damping with and
without exchange (programs I and 11) and the damping parameter
required to reproduce the experimental linewidth using
program I, assuming no pinning of the spins. If we assume a
surface an"isotropy of K = -0.1 erg/cm2 the calculated S
linewidths are approximately 50 Oe larger and the resonance
fields are approximately 50 Oe lower than if Ks = 0. The value
of the Gilbert damping parameter required to reproduce the (108)
linewidth with KS = -0.1 erg/cm2 was G = 6.6~10' sec-' as
opposed to G = 7.0x108 sec-' for KS = 0. The calculated
resonance fields are discussed in Section 4.6.
Experimental absorption derivative curves are shown in
Figure 4.4 for the three principal axes. These curves have been
TABLE 4-4
Results for 77 K, 23.95 GHz.
61001 [I101 [ I 1 1 1
Hfmr (kOe Sample 1 8.39r0.04 5.51 2.81
Sample 2 8.45 5.47 2.80
Calc. A 8.44 5.59 2.91
Calc. B 8.49 5.64 2.96
Calc. C 8.42 5.58 2.91
m(0e) Sample 1 880+50 1020 860
Sample 2 860 1020 840
G(sec-l) 7.Ox1O8 8.2x108 7.Ox1O8
Calc. A: No-exchange, no-damping value of Hfmr.
Calc. B: Local conductivity, no exchange, local damping with
the value of G listed in the Table, program 11.
Calc. C: Local conductivity, exchange, no surface anisotropy,
local damping with the value of G listed in the Table,
program I.
normalized to the same Pow field derivative peak amplitude. The
large shifts in the resonance due to MCA are evident by
comparing the field at which resonance occurs for the three
axes. The sharp spikes near zero field are due to domain wall
motion as at room temperature. The field values, 21~11/M~ and
I K I I/M~, at which the magnetization becomes parallel to the
applied field if the applied field is parallel to the (100) or
(110) axes respectively have been indicated on the figures.
Recall the calculated variation of the angle between the
magnetization and the applied field shown in Figure 2.7. The
irregular absorption below these field values is due to
absorption during rotation of the magnetization, see below.
Comparison of Experimental and Calculated Lineshapes
In Figure 4.5 we show a comparison of the experimental
(100) absorption derivative with calculated curves. The curve
in (a) was calculated with program I, using a value of
G = 7.0x108 sec-l, and no spin-pinning. The curve in (b) was
calculated using program I1 and the same value of G as was used
for the calculation shown in (a). The experimental asymmetry
differs from the calculated asymmetry, curve (a): however the
difference between calculation and experiment is smaller than at
room temperature. If we assumed a Gilbert damping of
G = 6.6x108 sec-' and a surface anisotropy Ks = -0.1 erg/cm2
instead of G = 7.0x108 sec-', the calculated lineshape was the
same as that shown in (a). Absorption derivatives were also
calculated using program 111, with a non-local conductivity and
a wavenumber independent damping. The lineshape obtained using
a non-local conductivity was almost indistinguishable from that
obtained using a local conductivity so the non-local line is not
reproduced here. Based on this comparison it appears that the
local conductivity theory may be used to describe the absorption
in Nickel at 77 K. From a comparison of the calculated
a I
(110 ABSORFTIQN
D E R I V A T I V E
(ARB. U N I T S )
0 5 10 1 9
I
0 5 10 I I
A P P L I E D F I E L D ( k O e )
Fiqure 4.4 Absorption derivatives vs applied field at 77 K,
23.95 GHz, Sample 2 for the three principal axes. The fields
I K I I / M ~ and 2 1 ~ 1 I / ~ ~ at which the magnetization becomes parallel
to the applied field if the applied field were parallel to the
(110) or (100) axes respectively are indicated on the
appropriate figure. 21~1]/~~=3.22 kOe at 77 K.
A B S O R P T I O N
D E R I V A T I V E
(ARB. UNITS)
APPLIED FIELD ( M e ) Figure 4.5 Comparison of calculated absorption derivatives with
experiment, 77 K t 23.95 GHz, Sample 2. The applied field was
parallel to a (100) axis. (a) Calculation using program I.
local conductivity. exchange, no surface anisotropy. A damping
parameter G = 7.0x108sec-' was used. Other parameters are
listed in Table 4-1. (b) Calculation using program 11, local
conductivity, no exchange, dragging. The parameters used in (a)
were used for this calculation.
linewidths for (a) and (b) it appears that exchange, with no
spin pinning, contributes approximately 80 Oe to the linewidth
at 77 K. Curve (b) demonstrates that the absorption at low
fields, (H,<~~K~I/M~), is due to dragging of the magnetization.
It is remarkable how well the simple model used for the dragging
processes describes the data. It is interesting to note the
discontinuity in the calculated derivative at 3.22 kOe where the
external field equals 2 1 ~ 1 1 / ~ ~ .
The Angular Variation of Hfmr
The variation of the resonance field with the orientation
of the applied field in the sample plane is shown in Figure 4.6.
for sample 2. A s was pointed out in Section 4.2 data can be
taken by rotating the magnet through only a limited range of
angles about the position where the applied field is parallel to
a principal axis and perpendicular to the microwave magnetic
field. As a result collecting the data shown in Figure 4.6
required cooling to 77 K three times, once for each axis, with a
return to room temperature to rotate the endwall assembly
holding the sample between each cooling. The solid line on the
figure represents the variation with angle expected on the basis
of program 11, which allows for arbitrary orientation of the
applied field with respect to the crystal axes. ragging of the
magnetization must bet taken into account when considering the
RESONANCE FIELD (kOe)
0 0 30 60 90
ANGLE (DEGREES)
Fiqure 4.6 Variation of the resonance field, Hfmr, with the
angle of the applied field in the sample plane, 77 K, 23.95 GHz.
Data taken by rotating about (100): 36 ; (11 I 1: t ; ( 1 10): o . Experimental uncertainty ( ~ 2 4 0 0e) is approximately the symbol
size. The solid line is the result of calculations made with
program 11, which allows for the lack of alignment between the
magnetization and the applied field.
angular variation of Hfmr and the linewidth since ZIKII/M~ is of
the order of Hfmr. To obtain these curves the absorption
derivative was calculated for each angle of the applied field
and the resonance field determined from the calculated curves.
Since this calculation did not include exchange we used a value
of G = 7.7x108 sec-' in order to reproduce the experimental
linewidth for the (100) direction. The experimental and.
calculated variations of Hfmr agree well.
Since the calculations were made using a value of G larger
than required by experiment, and neglecting exchange, the
calculated values of Hfmr will be larger than if the correct
value of G was used and exchange included, compare
calculations B and C in Table 4-4 for example. Because of the
neglect of exchange we would not expect the numerical values of
the calculated resonance fields to equal the experimental
values, however we would expect an approximately constant offset
between the experimental and the calculated values. The damping
is anisotropic so that the damping shift in Hfmr will be
anisotropic, but, as can be seen from Table 4-4, the anisotropy
in the shift is no greater than 10 Oe. Although the
experimental and calculated angular variations agree well there
. is a systematic difference between the offset for data taken by
rotating the magnet about the (100) axis and the data taken by
rotating about the ( 1 1 1 ) and (140) axes. I have no explanation
for this difference. It is clearly an experimental problem
because it does not show up in the room temperature (Figure 4.3)
or the 4.2 K (~igure 4.13) angular variations. The difference
is too large to be explained by a difference in the microwave
frequency used for the different sets of measurements. A
difference in the microwave frequency of 0.1 GHz (half of the
tuning range of the klystron) would shift the resonance field by
only 30 Oe.
The Angular Variation of AH
The variation of the linewidth with the direction of the
applied field in the sample plane is shown in Figure 4.7. The
linewidths plotted here were measured at the same time as the
resonance fields shown in Figure 4-6. The solid line in the
figure was calculated using the procedure outlined above for the
calculation of the angular variation of the resonance field.
This calculated angular variation of the linewidth demonstrates
clearly the effects of dragging on the FMR linewidth. Note
especially the large increase in the linewidth at angles near
the (100) and ( 1 1 1 ) directions. We do not expect the calculated
variation of the linewidth to reproduce the experimental
variation since an isotropic damping parameter was assumed for
the calculation with the result that the calculated linewidths
for the (loo), ( 1 1 1 ) and (110) directions were the same. The
discrepancy between experiment and calculation near the ( 1 10)
direction is certainly because the damping is not isotropic.
The discrepancy at angles between the (1 00) and (1 1 1 ) directions
'H (Oe)
0 30 60 90 ANGLE (DEGREES)
Fiqure 4.7 Variation of the FMR linewidth, AH, with the angle of
the applied field in the sample plane, 77 K, 23.95 GHz. Data
. taken by rotating about ( 1 0 0 ) : K ; 1 : + ; ( 1 1 0 ) : o . The solid line is the result of calculations made with program 11,
which allows for the lack of alignment between the magnetization
and the applied field. An isotropic damping parameter
G = 7.7x108sec-I was assumed for the calculation.
-167-
is possibly due to an anisotropy of the damping parameter.
4.4 Results at 4.2 K
Data at 4.2 K were collected using a bolometer to measure
the absorption rather than the absorption derivative as was
measured with the field modulation technique used at higher
temperatures. It was necessary to use the bolometer because the
FMR line becomes very broad and the field modulation system
lacked the sensitivity required to detect the FMR signal. The
result of an experiment was a signal proportional to the power
absorbed by the sample as a function of the applied field. This
data was stored on a computer and could be handled numerically.
For analysis the data was differentiated numerically and the
linewidth and resonance field obtained from the derivative.
The experimental values of the resonance field and the
linewidth for the two samples at 4.2 K, at 23.95 GHz are listed
in Table 4-5. The results for the two samples agree within the
experimental uncertainty. Our results indicate that the
linewidth is anisotropic at 4.2 K. As at 77 K the
(110) linewidth was greater than the (111) and (100) linewidths.
- The difference between the (110) and the (111) linewidths was
approximately 200 Oe or 10%. At 77 K we found that the (100)
and (111) Pinewidths were the same, however at 4.2 K the
(111) Pinewidth was greater than the (100) linewidth, by
approximately 200 Oe. The differences between the linewidths
-168-
TABLE 4-5
Results for 4.2 K, 23,95 GHz.
e1003 [1103 [ 1 1 1 1
Hfmr ( kOe) Sample 1 9.56k0.85 5.17 2.15
Sample 2 9.65 5.23 2.22
Calc. A 9.96 5.58 2.48
Calc. B 10.09 5.78 2.53
Calc. C 9.99 5.63 2.56
Calc. D 9.97 5.61 2.53
Calc. E 9.62 5.22 2.15
AH(Oe) Sample 1 1600k50 2000 1800
Sample 2 1640 2100 1830
Calc. A: No-exchange, no-damping values of Hfmr.
Calc. B: Local conductivity, no exchange, local damping, values
of G as listed above, program 11.
Calc. C: Local conductivity, exchange, local damping, values of
G as listed above, program 111.
Calc. D: Non-local conductivity, exchange, local damping, values
of G as listed above, program 111.
Calc. E: Non-local conductivity, exchange, isotropic non-local
damping, values of a=1.19x108sec-l, b=1.07~10~sec-~,
ID=620 A at 4.2 K (calculated linewidth at 4.2 K=1610 Oe),
program III.
for the three axes are four times the experimental uncertainty
in the linewidth, 250 Oe.
Also listed in Table 4-5 are values of Hfmr calculated with
a number of combinations of damping, exchange and conductivity:
the resonance fields for the (100) direction were calculated
using the value of K1' which includes the MCA torque ascribed to
the X, pocket[44]. If this contribution were neglected the
calculated resonance fields for the (100) direction would be
shifted 180 Oe to higher fields. The calculated values listed
in Table 4-5 will be discussed in Section 4.6, but it is worth
noting here the wide variations between them. In particular the
difference between the calculated resonance fields for the
wavenumber dependent and the wavenumber independent damping. E
and D respectively in the Table, is approximately 400 Oe. The
values of the damping parameter, G, listed in the Table are
those required to reproduce the experimental linewidths using
program 111 with a non-local conductivity and a wavenumber
independent damping. The surface anisotropy was assumed to be
zero in all of these calculations. The difference between the
linewidth calculated using program I1 with G = 14x108 sec-' and
that calculated using program 111 with a non-local conductivity
. and G = 14x108 sec-I was 30 Oe, so that exchange, with no spin
pinning, contributes approximately 30 Oe to the linewidth at
4.2 K. A surface anisotropy Ks = -0.1 erg/cm2 changes the
calculated linewidth by less than 10 0e and shifts the resonance
by approximately 25 Oe to lower fields. We may safely neglect
spin pinning in the discussion of our 4.2 K results which
follows.
The experimental absorption curves for the two samples at
23.95 GHz are shown in Figure 4.8. As at 77 K the irregular
absorption at fields below 2IK1I/MS or IKII/MS for the (100) and
(110) axes respectively occurs during rotation of the
magnetization. These field values have been indicated on the
figures. The experimental zero has been suppressed on these
figures, but except for the (100) trace of sample 2, the curves
have not been scaled relative to each other. The zero for the
(100) trace of sample 2 was mistakenly taken without shorting
the input to the lock-in amplifier, see'chapter 3, and so the
scaling of the data by the data acquisition program was
different to that of the other curves shown. By analogy with
Figure 4.8(a) this curve has been scaled to give the same
absorption at the peak and at the saturation field 21K11/Ms as
the (110) absorption curve for this sample.
In Figure 4.9(a) we show a comparison of the absorption
curves for the two samples. Since the experimental zero is not
meaningful the curves have been scaled vertically to the same
absorption at 200 Oe and at the peak for this comparison. The
agreement between the absorption lineshapes for the two samples
for the two other crystal axes was of similar quality. In
Figure 4.9(b) is a comparison of the absorption measured with
the bolometer and the absorption measured by monitoring the
microwave power reflected from the cavity with the microwave
ABSORPTION
(ARB. U N I T S )
APPLIED FIELD (kOe)
- Fiqure 4.8 ~bsorption curves at 4.2 K, 23.95 GHz for the three
- crystal axes. ( a ) Sample 1: (b) Sample 2; The fields IKII/MS and
~ I K I ~ / M , at which the magnetization becomes parallel to the
applied field if the applied field is parallel to the (110) or
(100) axes respectively are indicated on the appropriate figure.
2(~11/~~=4.92 kOe at 4.2-K.
ABSORPTION APPLIED FIELD (kOe )
APPLIED FIELD ( m e )
Figure 4.9 ( a ) Comparison of the absorption for sample 1 and
sample 2. The applied field was parallel to ( 1 1 1 ). The curves
have been scaled vertically to match at 200 Oe and at the peak.
(b) Comparison of the absorption measured using the bolometer
(solid line), and the microwave diode ( + ) , sample 2. The
applied field was parallel to (100).
diode. The diode data was taken by sweeping the applied field
in the direction of increasing and decreasing field and
averaging the two curves to compensate for an approximately
linear drift with time in the diode voltage. The lineshapes
measured with the two techniques agree well which gives some
confidence in the data obtained with the bolometer. Although
the diode data appears smooth in the figure we were unable to
obtain a reliable value of the linewidth by differentiating the
data numerically, even with considerable massaging before
differentiation.
In Figure 4.10 we show an absorption curve measured using
the bolometer, with the numerically calculated derivative
superimposed. The linewidth has been indicated on the figure to
indicate the position of the inflection points relative to the
absorption peak.
Comparison of Experimental and Calculated Lineshapes
In Figures 4.11 and 4.12 we show comparisons of calculated
curves with the experimental curves. The calculations shown
were made using program I11 with a non-local conductivity.
Since ql > 1 at FMR at 4.2 K, see Table 2-3, the wavenumber
dependence of the conductivity is important and must be
considered when calculating the absorption for comparison with
experiment. A wavenumber dependent (non-local) damping was
assumed for the calculation shown in Figure 4.11, and a
Fiqure 4.10 The absorption and the absorption derivative at
4.2 K, 23.95 GHz for sample 2. The applied field was parallel
to (100). The FMR linewidth is indicated on the figure.
wavenumber independent damping for the calculation shown in
Figure 4.12. The experimental data is the resonance for the
(100) direction of sample 2. The absorption was calculated
ignoring MCA and the resulting curves were shifted along the
field axis until the peak position matched that of experiment.
This is valid since MCA shifts the position of the resonance but
has no effect on the lineshape, at least for the (100) and (1 1 1 )
directions where the effective MCA fields, a and 7, are equal.
We consider in Section 4.6 the effect of the form of the damping
on the resonance field, but for the moment we consider only the
lineshape. The comparison between calculation and experiment is
shown for both the absorption and the absorption derivative. In
a comparison between the calculated and experimental derivatives
the attention is drawn to the field region around the FMR peak,
while in a comparison of the absorption attention is focussed on
the tails. It is worthwhile to examine both cases. The results
of the calculations are shown only for fields greater than
21K11/~~ as the calculation is not valid if the magnetization is
not parallel to the applied field.
The form of the wavenumber dependent damping assumed in the
calculation was:
The curve shown in Figure 4.11 was calculated using the values
ABSORPTION
DERIVATIVE
(ARB. U N I T S )
APPLIED FIEZIT) ( k O e )
Fiqure 4.11 Comparison of calculated absorption and absorption
- derivative with experiment, 4.2 K, 23.95 GHz, sample 2. The
applied field was parallel to (100). The calculation assumed a
non-local conductivity and a wavenumber dependent damping with
a = 1 . 0 7 ~ 1 0 ~ secml b = 1.19~10~ sec'l and I,, = 620 A. he
calculated curves have been shifted along the field axis so that
the FMR peak positions coincide.
ABSORPTION
(ARB, UNITS )
ABSORPTION
DERIVATIVE
(ARB. UNITS )
APPLIED FIELD (We )
Figure 4 . 1 2 Comparison of calculated absorption and absorption
derivative with experiment, 4 . 2 K, 2 3 . 9 5 GHz, sample 2 . The
applied field was paralle1,to ( 1 0 0 ) . The calculation assumed a
non-local conductivity and a wavenumber independent damping
F = 1 4 x 1 0 8 sec-'. The calculated curves have been shifted along
the field axis so that the PMR peak positions coincide.
a = 1.07x108 sec-l and b = 1.19x108 sec-I suggested by Cochran
and Heinrich[37], using the experimental resistivity ratib of
38, and the value of the d-electron mean free path, l D = 620 A
at 4.2 K t adjusted to reproduce the experimental linewidth of
1640 Oe. This 4.2 K mean free path corresponds to a room
temperature mean free path l D = 16 A. Calculations were made
with a variety of values of a, b and ID, subject to the -
constraints that the room temperature damping parameter,
essentially (a + b), be 2.45x108 sec", and that the 4.2 K
linewidth be 1640 Oe. The lineshapes calculated using a = 0.8,
1.2 and 2.0x108 sec-I with corresponding values of b and ID,
were virtually identical. The positions however were different,
see Section 4.6. The agreement between the calculated and
experimental lineshapes is superb if one looks only at the high
field side of the resonance, Ho > Hfmr. The lineshapes on the
low field side, Ho < Hfmr, do not agree at all.
In Figure 4.12 the comparison is made for a wavenumber
independent damping G = 14x108 sec-I. Again the calculated
curve has been shifted along the field axis to match the
experimental peak position. The agreement between the
calculated lineshape and the experimental lineshape is good in
the peak region but not so good in the tails. The agreement
between the experimental and calculated asymmetries is
excellent. This match is equivalent to that shown by Bhagat and
Hirstel] in their Figure 3. If fits to the lineshape were the
only consideration it would appear that the data was better
described by a wavenumber independent damping than by a
wavenumber dependent damping.
The Angular Variation of Hfmr
In Figure 4.13 we show the angular variation of the
resonance field. The solid line is the result of calculations
made using program PI, which neglects exchange, using an
isotropic wavenumber independent damping parameter
G = 14x108 sec-' to reproduce the (100) linewidth. The
experimental and calculated angular variations agree well
although there is an offset of approximately 400 Oe between - them. This is simply because the damping and exchange shifts
are not treated correctly in the theory.
FMR at 9.495 GHz at 4.2 K
Finally we present the results of measurements on sample 1
at 9.495 GHz at 4.2 K. The data is shown in Figure 4.14 for the
applied field along (100). The field 21Kll/MS is indicated on
. the figure. The subsidiary peak is a result of dragging. As a
result of MCA shifts and dragging FMR can be observed only when
the applied field is within a small angle of a (100) direction.
The signal amplitude decreased rapidly as the field was tilted
away from the (100) direction and disappeared entirely at an
lo 1 RESONANCE FIELD (kOe)
ANGLE (DEGREES)
Fiqure 4.13 Variation of the resonance field, Hfmr, with the
direction of the applied field in the sample plane, 4.2 K,
23.95 GHz. The data was taken by rotating the magnet about:
(100) * : ( 1 1 1 ) + . The experimental uncertainty is less than the symbol size. The solid line is calculated using
program PI with an isotropic damping parameter,
G = 14x108 see".
ABSORPTION
(ARB. UNITS )
APPLIED FIELD ( M e )
Fiqure 4.14 Absorption curve at 9.495 GHz, at 4.2 K, sample 1.
The applied field was parallel to the (100) axis. The field
- 2IK1I/MS at which the magnetization becomes parallel to the
applied field is indicated on the figure. The double peak is an
effect sf dragging of the magnetization.
angle of, very approximately, 10' between the field and the
(100) direction. This is also a result of the dragging of the
magnetization due to MCA, see the discussion in Section 2.3.
The resonance field and the linewidth for the main peak
were Hfmr = 5.76k0.05 kOe and AH = 790250 Oe. The values of the
resonance field and linewidth calculated using program I11 with
a non-local conductivity and a wavenumber dependent damping
using the parameters which described the 23.95 GHz linewidth,
a = 1.07x108 sec-I, b = 1.19x108 sec-I and ID = 620 A, were
5.80 kOe and 830 Oe respectively. These values are in good
agreement with the experimental values. The agreement between
lineshapes calculated assuming a wavenumber dependent or a
wavenumber independent damping and the main peak was similar to
that of the comparisons with the 23.95 GHz data shown in
Figures 4.11 and 4.12.
4.5 Results at Intermediate Temperatures
The Temperature Dependence of the Linewidth
The temperature dependence of the FMR linewidth for
sample 1 for the three axes is shown in Figure 4.15. We found
that the linewidths for the (100) and (111) directions for
temperatures greater than approximately 60 K were the same
within experimental uncertainty. As discussed in Section 4.4
the (111) linewidth at 4.2 K was larger than the (100)
TEMPERATURE (K)
Fiqure 4.15 The variation of the FMR linewidth, AH, with
temperature for the three principal crystal axes, 23.95 GHz,
(100) ; ( 1 1 1 ) ; (110) .The u n c e r t a i n t y i n t h e l i n e w i d t h
for temperatures below 100 K was f50 Oe. To avoid confusion in
the plot a single error bar is shown at 4.2 K. The solid line
is the data of Bhagat and ~ubitz[l3] scaled by the ratio of the
microwave frequencies, 23.95/22.
linewidth. The linewidth for the (110) direction was the same
as the (100) and ( 1 1 1 ) Pinewidths at room temperature but was
larger than the (100) and ( 9 1 1 ) linewidths at 200 K and below
for this sample.
The solid line in Figure 4.15 is the data of Bhagat and
Lubitz[l3] at 22 GHz, scaled by the ratio of our microwave
frequency to theirs, 23.95/22. The validity of this scaling is
discussed in Section 4.6. This data was taken from Figure 15 of
[ 131 (a larger version of Figure 5 of 1 1 2 1 ) . The data of Bhagat
and Lubitz matches our (100) and ( 1 1 1 ) data quite well at
temperatures above 60 K, and our (100) linewidth at 4.2 K, if
the difference in the microwave frequency is considered. We are
unable to say anything about the saturation of the linewidth
from our data, however the close correspondence with the data of
Bhagat and Lubitz is suggestive. The temperature variation of
the linewidth is discussed in Section 4.6 below.
The Temperature Dependence of Hfmr
The variation with temperature of the resonance field for
the three principal axes is shown in Figure 4.16. The solid
lines are the no-exchange no-damping values of Hfmr calculated
using the MCA constants of Tokunaga[67]. The inset to
Figure 4.16 is included to demonstrate the effect of the higher
order MCA constants on the resonance'position. Curves are shown
in the inset for a calculation made using only K1 and a
RES 0 IVA NC 2
FIELD (kOe)
FIELD
,100 ,200 I
TEN PERATURE
I I I 100 200 3 0 ;
TEV PERATURE (K )
Fiqure 4.16 The variation of the resonance field, Hfmr, with
temperature for the three principal crystal axes, 23.95 GHz.
The experimental uncertainty is indicated approximately by the
symbol size. The solid lines represent the no-exchange
- no-damping values of Hfmr calculated using K1, K2 and K3. The
inset shows the no-exchange no-damping values of Hfmr calculated
using K1 only compared with the calculation using K1, K2 and K3
to demonstrate the effect of the higher order MCA constants on
the resonance position.
calculation which used K1,
plot that the higher order
determining Hfmr in Nickel
calculated lines are shown
- 186-
K2 and K3. It is clear from this
MCA constants are important in
at low temperatures. These
because they are simple to calculate
and because they demonstrate, in a qualitative manner, the
temperature dependence which may be expected for Hfmr.
Tokunaga's MCA constants are used because they are available for
the whole temperature range, room temperature to 4.2 K. As
pointed out in Section 4.1 there is a discrepancy between the
higher order MCA constants of Tokunaga and those of
Tung et a1[21] at the temperatures at which they can be
compared. Because of the uncertainty in the values of the MCA
constants it is difficult to extract any information from this
data. If we wished to determine values for the MCA constants
from this data we would have to know the damping and exchange
shifts. Conversely if we wished to determine the damping and
exchange shift we would need to know the MCA constants
accurately. We know the MCA constants at 77 and 4.2 K well and
so are restricted to those temperatures for an analysis of the
damping and exchange shifts. These are discussed in
Section 4.6.
4.6 Discussion
In this Section we discuss the results which have been
presented thus far in this Chapter. First we discuss our
linewidth data, how it compares with that of other workers and
the implications of our measurements. Then we discuss the
consequences of a wavenumber dependent damping of the form of
equation ( 4 . 3 ) for the temperature dependence of the linewidth
and the resonance field.
Let us start by summarizing our linewidth results. At room
temperature and for 23.95 GHz we found that the linewidths were
an average of 40 Oe larger than the ideal linewidth of 320 Oe.
The linewidth for sample -1 was isotropic within 15 Oe, while
there was a 40 Oe spread in the linewidths for the different
crystal axes for sample 2. The frequency dependence of the
linewidth for this sample showed that the ( 1 1 0 ) linewidths were
consistently larger than the ( 1 0 0 ) and ( 1 1 1 ) linewidths by an
amount roughly equal to the experimental uncertainty. The
frequency dependence of the linewidth was consistent with a
surface anisotropy of KS = -0.1 erg/cmz.
The ( 1 10) linewidth for sample 1 was larger than the ( 1 0 0 )
and ( 1 1 1 ) linewidths for this sample at 200 K, the highest
temperature measured below room temperature. The ( 1 0 0 ) and
( 1 1 1 ) linewidths were the same at all temperatures above 60 K.
At 77 K the ( 1 1 0 ) linewidth for both samples was approximately
16% larger than the ( 1 0 0 ) and ( 1 1 1 ) linewidths at that
temperature. At 4.2 K the (110) linewidth was 2050250 Oe, the
( 1 1 1 ) linewidth was 1815+50 Oe and the (100) linewidth was
1620+50 Oe. The linewidths for the two samples were in good
agreement at all temperatures.
The low temperature data available for comparison includes
the measurement of ~ranse[28] at 77 K, the measurements of
Anders, Bastian and Biller[l7] at temperatures from 77 K.to
630 K and the measurements of Bhagat and ~irst[t] and Bhagat and
~ubitz[l2,13] at temperatures from 4.2 K to room temperature.
~ranse[28] measured a linewidth of 1200 Oe at 23.3 GHz at
77 K. He does not state the orientation of the magnetic field
for this measurement. Franse's room temperature linewidth was
600 Oe or approximately twice the linewidth due to the intrinsic
damping and exchange conductivity. Presumably part of his large
77 K linewidth was due to the increase in the intrinsic damping
and part was due to whatever was responsible for the extra
linewidth at room temperature. This linewidth is larger than
our (110) linewidths at this temperature, 1020 Oe, and our (100)
and ( 1 1 1 ) linewidths, 860 Oe.
Anders et al[17] made measurements on carefully annealed
and electropolished (110) Nickel disks. They measured the
linewidth for the three principal crystal directions at 9.19,
19.67 and 26.2 GHz at temperatures from 77 K to 630 K. Their
room temperature lines were'narrow, being 350 Oe at 26.2 GHz,
the ideal linewidth at this frequency. The room temperature
linewidths were isotropic within a spread of approximately 50 Oe
at 26.2 GHz. They found that the (110) linewidth became larger
than the (100) and the ( 1 1 1 ) linewidths at temperatures below
273 K, and that the difference in the linewidths increased with
decreasing temperature. At 77 K they had the (110) linewidth
equal to 820 Oe, and the ( 1 1 1 ) linewidth equal to 640 Oe at
26.2 GHz. No value for the (100) linewidth is quoted at this
frequency but it appears from the data for the other frequencies
that there was no significant difference between the ( 1 1 1 ) and
(100) linewidths. Our observations as to the anisotropy of the
linewidth are in agreement with Anders et al. Their 77 K
linewidths are much narrower than those measured by us and by
Bhagat and Lubitz[l2,13]. They do not quote a resistivity ratio
for their samples. It ism likely that their Nickel was less pure
than ours or that of Bhagat and Lubitz. Since the linewidth
increases with increasing resistivity ratio the linewidth for a
lower purity sample should increase less rapidly with decreasing
temperature than the linewidth for a pure sample, Recall that
Lloyd and Bhagat[l4] found no increase with decreasing
temperature in a 5.4% Copper in Nickel alloy.
Bhagat and Hirst report measurements made on cylinders
oriented with a (100) or a ( 1 1 1 ) direction parallel to the
cylinder axis and on (190) disks. The orientation of the
applied field in the sample plane for the disk measurements is
not stated. Presumably these authors did not make any
measurements with the applied field along the (110) direction
and so make no comment as to an anisotropy of the linewidth for
this direction as compared with the (100) or (111) directions.
They make no mention either of a difference between the (100)
and the (111) linewidths at 4.2 K.
Bhagat and Lubitz report measurements on (111) cylinders.
To compare our data with that of Bhagat and Lubitz the simplest
thing to do is to simply multiply their linewidths by the ratio
of the microwave frequencies, 23.95/22. In doing this we ignore
the zero frequency intercept in the frequency dependence due to
exchange. The exchange contribution to the linewidth is small,
and the frequencies are quite close, so the error introduced
thereby is negligible. For example at 77 K the error is less
than 10 Oe. At 77 K and 4.2 K Bhagat and Lubitz have linewidths
of 780 Oe and 1480 Oe respectively. When scaled by the ratio of
the frequencies these linewidths become 850 Oe and 1610 Oe
respectively, which are in good agreement with our (100) and
(111) linewidths at 77 K, 860 Oe, and with our (100) linewidth
at 4.2 K, 1620 Oe. Our (111) linewidth at 4.2 K, 1815 Oe, is
larger than that of Bhagat and Lubitz.
We need to ask what else besides an anisotropy in the
intrinsic damping could produce the observed anisotropy in the
linewidth, especially the difference between the (100) and ( 1 1 1 )
linewidths at 4.2 K. It is unlikely that it could be due to any
strain in the surface due to the surface preparation or to
strain in the sample induced by the mounting used since the
anisotropy for the (111) and (100) directions appears only at
temperatures below 60 K. If the anisotropy were produced by
strain it would be expected that the (100) axis, being the hard
MCA axis, would be affected more than the ( 1 1 1 ) axis, which is
the easy MCA axis, with the result that the (100) linewidth
would be greater than the ( 1 1 1 ) linewidth because of effects due
to the misalignment between the magnetization and the applied
field. I can think of no experimental factors which would
produce a temperature dependence of the (100) and ( 1 1 1 ) .
linewidths similar to that which we have observed.
With the qualification that the measurements were made on
samples cut from the same boule, so that the effect may be a
result of a peculiarity of the sample, we conclude that the
effect is real and is due to an anisotropy of the damping
parameter. The disagreement between our ( 1 1 1 ) linewidths and
those of Bhagat and Lubitz remains unexplained. It would be
worthwhile to repeat our measurements on samples cut from a
different single crystal to be absolutely sure that the
difference between the (100) and ( 1 1 1 ) linewidths at 4.2 K is
not a sample dependent effect. In any event the anisotropy for
the (110) linewidth appears well established since it has been
observed by both Anders, Bastian and Biller and by us.
The Wavenumber Dependent Damping
We now wish to examine the consequences of a wavenumber
dependent damping of the form:
We will discuss the temperature dependence of the linew.idth, AH,
and the shift in the position of the resonance, 6M, defined as
the difference between the value of Hfmr expected using (4.3)
and the no-exchange no-damping value of Hfmr. It is
straightforward to compare the calculated temperature dependence
of the linewidth with experiment, however it is difficult to
compare the shift, 6H, because the resonance is also shifted by
MCA. The MCA shifts are much larger than the damping and
exchange shifts, 6H. For example at 4.2 K the MCA shift for the
( 1 0 0 ) direction, 21Kll/Ms-, is 4.92 kOe while the damping and
exchange shift is of the order of 300 Oe, from the numbers in
Table 4-5. To compare the calculated shifts with experiment we
would need to know the MCA shifts accurately. Conversely, to
determine the MCA constants from our data we would need to know
the damping and exchange shift.
Cochran and HeinrichL371 fitted the temperature dependence
of the damping parameter deduced from FMAR transmission
experiments with the limiting form of (4.3) for small q:
Their values of a and b were a = 1.07x108 sec" and
b = 1.19x108 sec''. Rather than trying to fit the temperature
dependence of our linewidths we will display some representative
possible calculated temperature dependences. A t room
temperature (295 K) ql << 1 so that:
Taking the value of G = 2.45x108 sec-l at room temperature, we
have a constraint on the values of a and b which may be used in
(4.31, i.e. (a + b) = 2.45x10a sec-I. We impose as a second
constraint on the parameters entering (4.3) that the calculated
linewidth equal the average of our (100) linewidth at
4.2 K, 1620 Oe. So for a given value of a the values of b and
of I D at any temperature-are fixed. We have chosen values of
a = 0.8, 1.2, and 2.0x108 sec-' as covering a wide range of
ratios of intra-band to inter-band scattering at room
temperature. The values of b, and of I D at room temperature,
corresponding to these values of a were b = 1.65, 1.25, and
0.45x108 sec-l, and I D = 10, 16, and 28 A respectively.
In figure 4.17 the linewidth calculated using program I11
with a non-local conductivity and a wavenumber dependent damping
with the three sets of parameters a, b, and I D is plotted as a
function of the logarithm of the resistivity ratio p(~)/p(295).
This is a convenient way of displaying the results since it is
the resistivity ratio which enters the damping (4.3). Also
shown on the figure is our data for the temperature dependence
of the (100) linewidth. Resistivity ratios of 10 and 38 are
LINEWIDTH (Oe)
I
1 10 38 100 RESISTIVITY RATIO
Fiqure 4.17 The variation of the FMR linewidth with the
logarithm of the resistivity ratio. The solid lines were
calculated with program I11 with a non-local conductivity and a
wavenumber dependent damping with (A)a = 0.8x108 sec-l,
b = 1.65x108 sec-l; (B)a = 1.2x108 sec-l, b = 1.25x108 sec-':
(C)a = 2.0x108 sec'l, b = 0.45x108 sec-l; The circles are the
experimental data. Resistivity ratios of 10 and 38 correspond
to temperatures of 77 and 4.2 K respectively for our samples.
indicated on the figure, corresponding to 77 and 4.2 R
respectively for our samples. A temperature was associated with
each resistivity ratio using the resistivity ratios of our
samples. The value of the magnetization corresponding to that
temperature was used in the calculations. The magnetization is
not a strong function of temperature in the temperature range we
are considering so the effects of a small inaccuracy in relating
the temperature to the resistivity ratio should be negligible.
The calculated variation of AH with temperature
(resistivity ratio) exhibits the increase with decreasing
temperature (increasing resistivity ratio) observed
experimentally and the saturation at large resistivity ratios
discussed by Bhagat and ~irst[l]. As expected the linewidth
saturates at higher temperatures for larger values of a, that is
for a larger contribution of the intra-band damping to the total
damping at room temperature. For the smallest value of a shown
the linewidth had not saturated at a resistivity ratio of 38.
The values of AH at saturation were AH = 1770, 1700 and 1630 Oe
respectively for a = 0.8, 1.2, and 2.0x108 sec-l.
Comparing our data with these calculated curves it appears
that the data follows the temperature dependence calculated
assuming a = 0.8x108 sec"' reasonably well. Of course there is
the problem of the extra linebroadening in our experimental
results, but this would not affect our linewidths by more than
approximately 40 Oe at any temperature, see the discussion's in
Sections 4.2, 4.3 and 4.4. The values of a and b which would be
chosen to match the experimental temperature dependence would be
close to a = 0.8x108 sec-' and b = 1.65x108 sec-'. It would be
possible to better define the best values of a and b, but it is
probably not worth the large effort.
In Figure 4.18 the damping and exchange shift 6H, is
plotted as a function of the logarithm of the resistivity ratio
for the three sets of a, b and I D . As in Figure 4.17 resistivity
ratios of 10 and 38 have been indicated on the figure. The
variation of 6H for the three sets of parameters are quite
similar, the damping and exchange shift being to lower fields
(6H is negative). At a resistivity ratio of 38 the shifts are
6H = -200, -280, and -340 Oe for a = 0.8, 1.2, and 2.0~10~' sec-l
respectively.
Before attempting to compare these calculated temperature
variations with experiment it is instructive to examine the
temperature variation of 6H to be expected for a wavenumber
independent damping. In Figure 4.19 we have assembled the
results of calculations for the following combinations of
damping and conductivity:
(a) non-local conductivity, wavenumber dependent damping. For
this plot we have used the parameters of Cochran and Heinrich
with I D = 16 A at room temperature as in Section 4.4. The
variation of 6H is similar to that calculated assuming
a = 4.2x108 sec-I in Figure 4.98.
(b) non-local conductivity, wavenumber independent damping
G = 2.45x108 secPf. This is the variation of 6H which would be
1 10 38 100
RESISTIVITY RATIO
Figure 4.18 The variation of the shift, 6H, with the logarithm
of the resistivity ratio. The shift is defined as the
difference between the no-exchange no-damping value of Hfmr and
that calculated with program 111. A non-local conductivity and
a wavenumber independent damping was assumed, with
(A)a = 0.8x108 sec-l, b = 1.65x1Q8 sec": (B)a = 1.2x108 sec-I,
b = 1.25x108 secel; (C)a = 2.0x1Q8 sec'l, b = 0.45x1Q8 sec-':
Resistivity ratios of 10 and 38 correspond to temperatures of 77
and 4.2 K respectively for our samples.
RESISTIVITY RATIO ,
Fiqure 4.19 The variation of the shift, 6H, with the logarithm
of the resistivity ratio. The curves were calculated assuming:
(a) non-local conductivity, wavenumber dependent damping;
(b) non-local conductivity, wavenumber independent damping
G = 2.45~10' sec-'; (c) non-local conductivity, wavenumber
independent damping G = 8.0~10' sec-l; (dl non-local
. conductivity, wavenumber independent damping G = 14x10' sec-';
(el local conductivity, wavenumber independent damping
G = 2.45x108 sec-I. The circles are the experimental data at 77
and 4.2 K. ~esisfivity ratios of 10 and 38 correspond to
temperatures of 77 and 4.2 K respectively for our samples.
expected for a material having a temperature independent damping
equal to that of Nickel at room temperature.
(c) the same as (b) except that G = 8.0x108 sec-', approximately
the damping parameter required to reproduce the linewidth in
Nickel at 77 K.
(dl The same as (b) except that G = 14x108 sec'l, the damping
parameter required to reproduce the (100) linewidth in Nickel at
(el to demonstrate the effect of a local vs a non-local
conductivity curve (el has been calculated assuming a local
conductivity with G = 2.45x108 sec-l. This curve is useful
because it gives a rough idea of the temperature at which the
effects of a non-local conductivity become important, curves (b)
and (el diverge at a resistivity ratio of approximately 20.
The crosses on the Figure are the experimental shifts at 77 and
4.2 K calculated using the MCA constants of Tung et a1[21].
These shifts were obtained by subtracting Calc. A in Tables 4-4
and 4-5 from the experimental values of Hfmr.
The magnetic damping in Nickel is temperature dependent. A
feeling for the temperature variation of 6H if a wavenumber
independent damping were assumed can be obtained by looking at
curve (b) at room temperature, curve (c) at 77 K (resistivity
ratio = 10) and curve (dl at 4.2 K (resistivity ratio = 38).
The shift is small and not strongly dependent on temperature.
On the other hand the shift due to the wavenumber dependent
damping, curve (a), is strongly temperature dependent and much
larger than for a wavenumber independent damping. We may
compare these calculated shifts with experiment at 77 K and
4.2 K where the MCA shifts are known with reasonable certainty.
From Table 4-4 the shift 6M at 7 7 K varies from +10 to -120 Oe.
From Table 4-5 the shift at 4.2 K varies from -260 to -430 Oe.
At 7 7 K the difference between the calculated shifts for a
wavenumber dependent and a wavenumber independent damping are
small so that it is not possible to choose between the two forms
of the damping from the experimental values. However at 4.2 K
the wavenumber dependent damping shift is in much better
agreement with experiment than the wavenumber independent
damping shift, as evidenced by the data on the figure.
Rather than comparing the calculated shifts with experiment
we may approach the problem from a different direction and ask
how the values of the MCA constants deduced from experiment
assuming the two different forms of the damping compare with
accepted values. If a wavenumber dependent damping was assumed
the value of K1' obtained from the position of the resonance for
the (100) direction would be in good agreement with the value of
Tung, Said and Everettf211, Kf' = -12.44x105 erg.cm3. If a
wavenumber independent damping was assumed the value of K1'
would be K1' = -11.5x105 erg.cmJ. To demonstrate that our
results are not a peculiarity of our samples we cite the value
of IKII/M~ of 2150 G quoted by Lloyd and ~hagatfl41 at 4.2 K.
This corresponds to a value ~ 1 ' = -11.3x1Q5 erg/cm3, in good
agreement with our wavenumber independent damping value of K1'.
To summarize, the temperature variation of the (100)
linewidth was consistent with a wavenumber dependent damping of
the form (4.3) with a = 0.8x108 sec-l, b = 1.65x108 sec-' and
I D = 28 A at room temperature. The damping and exchange shift,
6H, also appears consistent with this form of the damping.
However the experimental lineshapes and those calculated
assuming a wavenumber dependent damping are only in partial
agreement, see Figure 4.11.
5 . CALCULATION OF THE DAMPING PARAMETER
5.1 Introduction
We now turn from experiment to a consideration of the
microscopic origins of magnetic damping. The first part of this
chapter contains a qualitative discussion of the effects of
spin-orbit coupling on electron states and how spin-orbit
coupling may lead to magnetic damping. This is followed by
presentation of a calculation of the damping parameter using a '
simple model of electrons and spin waves coupled through the
spin-orbit interaction. The ideas discussed here are largely
due to Kambersky[2,70,71,72]. The low temperature damping
mechanism has been discussed by Korenman and ~range[3,4,73].
~erger[74] has also presented a theory of magnetic damping
applicable to Nickel.
Spin-orbit coupling has two effects on the electron states
in a solid, it mixes the spin and it shifts the energy. These
two effects lead to two magnetic damping mechanisms with
different temperature dependences.
In the absence of spin-orbit coupling a band state is
either spin-up or spin-down. In the presence of spin-orbit
coupling the band states are not spin eigenstates. A state
Ik,n,+> where k is the momentum, n the band index and + the spin
index, becomes (following ~lliott[tO]) (akn+l+> + bkn+l->) and a
state lk,n,-> becomes (akn-I-> + b kn- I+>) where la1 is >> (bl.
The constants a and b depend on both k and n. Scattering of an
electron by a phonon or impurity results in a change of the spin
of the system. Three types of scattering may be distinguished
depending on whether the band and spin indices change: (i) an
electron in state Ik,n,+> scatters to Ik',n,+> (intraband
scattering), (ii) lk,n,+> scatters to Ik',n',+> (interband
scattering with no change of spin index), and (iii) lk,n,+>
scatters to (k',n',->. The third type of scattering (spin-flip
scattering) is not possible in the absence of spin-orbit
coupling. since the spin of the system is not conserved it is
clear that scattering may lead to magnetic damping. Apparently
spin-flip scattering gives the largest contribution to the
magnetic damping. The magnitude of the damping depends on the
relative magnitudes of the gap between the bands (AE) and the
reciprocal lifetime of the electrons K/r. If a single gap is
present the damping varies as:
If K / r is <c AE the damping varies as 1/r, This has been
demonstrated by Heinrich, Fraitova and ~ambersky[75] who
considered the damping introduced by the s-electron d-electron
exchange interaction. In a real metal there is a spectrum of
energy gaps present ranging from zero at accidental degeneracies
to the full exchange splitting. The damping will consist of a
sum of terms like (5.1). It is thought that this mechanism is
responsible for the flat temperature dependence of the damping
in Nickel between approximately 200 K and 600 K. This mechanism
was considered in the damping used in Chapter 4 by the term that
varied as the resistivity. For pure metals T becomes large at
low temperatures so that the damping due to this mechanism
becomes small.
The other effect of spin-orbit coupling is to shift the
energy of an electronic state. This leads to a magnetic damping
responsible for a linewidth which has a temperature dependence
similar to that observed in Nickel at low temperatures, namely
an increase with decreasing temperature leading to saturation at
very low temperatures. In a ferromagnetic metal the shifts
depend on the direction of the magnetization. Generally the
shifts are small, being second order in the spin-orbit coupling
parameter 4121 which is small (4 for Nickel is of the order of
0.1 eV[161). The situation may be quite different if there are
degenerate states whose degeneracy is lifted by spin-orbit
coupling. The splitting of the bands then depends on the
direction of the magnetization with respect to the crystal axes,
. If the degeneracy is near the Fermi surface the shifts in the
energy levels lead to changes in the size and shape of the Fermi
surface. The effect of spin orbit coupling on the band
structure when degeneracies are present has been discussed by
~lliott[lO]. His Figures 3-6 demonstrate the effects which may
occur.
A useful picture, and the one which will be used in the
calculation of the damping parameter, is to consider the metal
as containing collective magnetic excitations (spin waves) and
single particle excitations (electrons). Precessional motion of
the magnetization may be described in terms of spin wave
amplitudes. FMR consists of exciting spin waves by the .
microwave magnetic field. The spin waves may be described in
terms of electron states but such a description need not concern
us. The electron and spin wave systems are coupled by spin
orbit coupling because the electron energy depends on the
direction of the magnetization of the magnetization. Magnetic
damping occurs when a spin wave is annihilated in a collision
with an electron and the electron is excited into a higher
energy state. Energy and momentum must be conserved in such a
collision. An estimate of the spin wave energy, momentum and
velocity is:
where o is the spin wave frequency (the microwave frequency) and
6 is the microwave skin depth (see Chapter 2). An estimate of
the electron energy, momentum and velocity is:
where a is the lattice spacing. Clearly Esw << E, q << k and
v << v If an electron is excited from a state E to a state sw F*
E' in a collision with a spin wave, then, by conservation of
energy and momentum we have:
Combining these two equations:
E' - E = (K2/2m)(2E*G + q2) = Kw
or:
where E = E2k2/2m, and qS has been neglected compared with 2E*GO
vFq is approximately 1 0 ' while o is approximately 1 0 ' SO that:
In other words only those electrons whose velocity is
approximately perpendicular to that of the spin wave interact
with the spin wave. More precisely the component of the
electron's velocity parallel to 6 must equal the spin wave phase velocity for an electron spin wave collision to occur. If the
electron lifetime r , due to phonon and impurity scattering, is
short the electron momentum is not well defined and'the momentum
conservation condition is not stringent. The number of
electrons which may interact with the spin wave is large but the
interaction time is short so that the damping is small. At low
temperatures where the lifetime increases the momentum
conservation condition is stringent and restricts the number of
electrons which may interact with the spin wave. However
because the lifetime is long the interaction is much more
effective and the total effect becomes large. We may think of
the electrons as 'surf-riding' on the spin wave. The energy
which the electrons may absorb from the spin waves is limited
only by the time of the ride. This leads to a damping which
increases with the electron relaxation time as is observed in
Nic kel.
The calculation which is carried out in this chapter is
based on this idea. The X, hole pockets in the Fermi surface of
Nickel are known to change size and shape with the direction of
the magnetization. We consider only those electrons in states
near these pockets. These are minority spin electrons so we
consider' only electrons of a single spin. We use the Fermi
surface of Hodges, Stone and Gold[l6], the description of spin
waves given by Sparks[6] and the variation of energy levels with
the direction of the magnetization given by Gold[76]. These are
described in Section 5.2. The approach is to calculate the
response of the spin wave system to a magnetic field which
varies as exp(i (qy-ot)) using the method of Green's functions.
This gives the frequency and wavenumber dependent susceptibilty.
The imaginary part of this susceptibility is related to the
damping parameter. This calculation is presented in
Section 5.3. The integrals over the Fermi surface which enter
the damping parameter have been evaluated. The results are
compared with the calculations of Kambersky and with the
experimental results presented in Chapter 4 in Section 5.4. Our
calculation is similar to that carried out by Heinrich, Fraitova
and Kambersky[75].
5.2 The Model
The Fermi Surface of Nickel
The band structure and Fermi surface of Nickel have been
calculated by a number of workers. The calculations of
~ornberg[77] are useful for the complete Fermi surface including
the effects of spin-orbit coupling. A recent reference is the
work of Weling and Callaway[781. A schematic sketch of the band
structure of Nickel as presented by Gold[69] is shown in
Figure 5.1, as well as the band structure near the X points, see
below.
Most people in the field seem to agree as to the large
features of the ~ermi surface although there are a few small
areas whose existence is still a matter of discussion. The
large features include six distinct sheets. There are two
sheets of predominantly s-character, a majority spin sheet and a
minority spin sheet. The two sheets are similar in shape having
pronounced 'bulges' in the ( 1 1 1 ) directions. The majority spin
sheet contacts the Brillouin zone edge in the ( 1 1 1 ) directions
with the formation of 'necks' similar to those of the Fermi
surface of copper. There is a predominantly d-character
minority spin sheet with 'bulges' in the (110) directions.
These three sheets are centered in the Brillouin zone. There
are three minority spin sheets centered about the X-points of
the Brillouin zone (the X-points are located at 2n/a(fl,O,O),
2n/a(O,fl,O) and 2a/a(O,O,fl)). These sheets arise from the X5
d-band. They are much smaller than the other three sheets
above, they are approximately ellipsoidal in shape and have
hole-like character. These are the X, hole pockets which form
the focus of this chapter. There is a possibility that small
hole pockets at the X-points arising from the minority spin X2
level may exist. The X2 level is close to the Fermi level and
may be shifted, in a calculation, above or below with small
0590 2.
- (?1.0.0) r-x-w r-- X -W
Fiqure 5.1 (a)A schematic sketch of the energy bands of Nickel
as presented by Gold[69].
(b)The band structure at the Fermi level near the X points as
given by Hodges, Stone and Gold[16]. The magnetization is
parallel to [001]. Solid curve: 5 = 0.1 eV; Dashed
curve: 5 = 0.
Inset Directions in the reciprocal lattice of a face. centered
cubic lattice.
changes in parameters, see Figure 5.1 where the X, level appears
just below the minority spin Fermi energy. The only
experimental evidence for the existence of these pockets is the
torque measurements of GersdorfL441 and Tung et a1[211.
According to Zornberg degeneracies in the band structure
occur near the L-points ( ( 1 1 1 ) directions), along A (r - L), along A (I' - X) and at accidental degeneracies due to band crossings which occur when the exchange splitting is added to
the band calculation. The band structure in the rest of the
Brillouin zone is largely independent of magnetic field
direction.
In the absence of spin-orbit coupling the X, 1eve.l is
doubly degenerate. The degeneracy is lifted by spin-orbit
coupling, the splitting of the levels depending on the angle
between the magnetization and the (100) axis of interest. Since
the position in k-space where the band crosses the Fermi level
changes with the direction of the magnetization, the dimensions
of the Fermi surface change.
This change in size and shape of the Fermi surface with the
direction of the magnetization was first invoked to explain
unusual de Haas-van Alphen results (~odges,Stone and Gold[16],
this paper will be referred to as HSG). These authors produced
a band structure and a F'ermi surface using the interpolation
scheme of Hodges, Ehrenreich and ~ang[79] which fitted the dHvA
data from the pockets well. The calculated band structure did
not fit the results for the rest of the Brillouin zone well.
However since we are interested only in the hole pockets we will
use the ~ermi surface parameters of HSG.
Dimensions of the hole pockets at the different X-points
for the magnetization along [001], [ 1 1 1 ] and [110] are listed in
Table 5-1. These numbers are taken from Table 1 of HSG. Also
listed in Table 5-1 are values of the spin-orbit coupling
parameter l , the Fermi energy EF and the energy of the X, level
at the X-point in the absence of spin-orbit coupling, Ex, quoted * *
by HSG. Effective masses m, for the direction kXW and m 2 for
the direction kXr in the absence of spin-orbit coupling are *
listed, as well as the Fermi velocity appropriate for EF and m,.
The band structure at the Fermi energy near the X-points is
shown in Figure 5.1 and the X, pockets are illustrated in
Figure 5.2 when MS points along [001]. The pockets are shown in
the presence and absence of spin-orbit coupling. Different
authors quote different dimensions for the pockets. For example
Weling and Callaway quote values of kXr ranging from 0.195 to
0.256 times 27r/a in the absence of spin-orbit coupling. A
comprehensive discussion of the pockets is given by
Zornberg[77].
To calculate the damping parameter we need to know the
dimensions of the Fermi surface and how the energies depend on
the direction of the magnetization. Gold[76] has given a simple
analysis of the dependence of the energy levels on the direction
of the magnetization. By considering only the degenerate levels
and neglecting any mixing from other states at the X-points and
-213-
TABLE 5-1
Distances from X to the surface of the hole pocket in units sf
2r/a, where a=3.5166 A (Hodges, Stone and Gold)
E = 0.1 eV
Field Location k ~ r kxw k~~
Direction of pockets
[0013 (o,o,+l) 0.195 0.100 0.094
(kl ,OIo) 0.220 0.142 0.108
(0,91 ,O)
and Parameters for 5 = 0
m = free electron mass
k vF(m,) = 5.6~10' cm/sec
Fiqure 5.2 The X, hole pockets in the Fermi Surface of Nickel,
based on the parameters of HSG. In this plot the pockets at
2n/a(1,0,0) and 2n/a(0,0,1) are shown. The magnetization points
along [ 0 0 1 ] . The dotted curves represent the Fermi Surface in
the absence of spin-orbit coupling. The solid curves represent
the Fermi Surface with t = 0.1 eV. The boundary of the
Brillouin zone is shown,
treating spin-orbit coupling as a perturbation he found that:
where E,(k) is the energy in the absence of spin-orbit coupling,
6 is the spin-orbit coupling parameter and OM is the angle
between the magnetization and the (1 00) axis being considered.
According to this picture if the magnetization is along [001]
the levels at [100] and [010] should not be shifted. This is
not true as can be seen from Table 5-1 and Figure 5.2. The
difference is small however and the expression (5.2) will be
used in the discussion which follows.
Neglecting the light fluting of the hole pockets, ie
considering them as ellipsoids with major axis kXr, and minor
axis kXW, the energy of electrons near the X-points may be
written:
where k is measured from the X-point and the kZ axis is along
the I?-X axis of the pocket being considered. These simplified
pockets change size, but not shape, with changes in the
direction of the magnetization.
Geometry for the Calculation
We now address the problem of actually calculating the
damping parameter for the low temperature damping mechanism.
The approach is to evaluate the microwave susceptibility using
the method of Green's functions (see below). The imaginary part
of the susceptibilty is related to the damping parameter'.
The geometry assumed is shown in Figure 5.3. The sample
forms a slab of infinite extent lying in the x-z plane. The
external field and the magnetization point along the
z-direction. We consider only cases where a principal crystal
axis is parallel to z . As was demonstrated in Section 2.3 the
magnetization will then be parallel to the applied field if the
magnitude of the applied field is greater than some critical
value. Microwaves travel in the +y-direction with the microwave
magnetic field in the x-direction. The time and space variation
exp(i (qy-wt)) is assumed. This geometry is essentially the same
as that of the calculations outlined in Chapter 2, Sections 2 . 1
and 2.3, however the coordinate system has been changed so that
the magnetization points in the z-direction (for quantum
mechanical reasons). Only small deviations of the magnetization
from equilibrium are considered. The components of the
magnetization are (mx,m ,M ) to first order in the small Y S
quantities mx and m ye
There are three principal crystal axes ( 100), ( 1 1 0) and
(111). In our experiments the samples were cut with a
[ 170 1 axis normal to the sample plane. We could measure FMR
with the applied field parallel to [0011, [ 1 1 0 ] or [ 1 1 1 ] with
the spin wave wavevector q along [ i 1 0 ] . Experiments may also be
performed on samples cut with an [ 0 1 0 ] axis normal to the sample
plane. The [ 0 1 0 ] plane contains the [ 0 0 1 ] and [ 1 0 1 ] crystal
axes. An experiment performed with a [ 0 1 0 ] normal sample with
the applied field along the [ 0 0 1 1 axis is not equivalent, to an
experiment performed with a [ 1701 normal sample and the applied
field along [ 0 0 1 ] as the direction of the spin wave wavevector
with respect to the crystal axes is different. Thus there are
five orientations of the crystal axes of interest: with the
sample plane being a [ 0 1 0 1 normal crystal plane, (i) Ms parallel
to the [ 0 0 1 ] axis, (ii) MS parallel to the [ 1 0 1 ] axis; with the
sample plane being a [ 1701 normal crystal plane, (iii) M S
parallel to the [ 0 0 1 ] axis, (iv) MS parallel to the [ 1 1 1 ] axis
and (v) Ms parallel to the [ 1 1 0 1 axis. Case (i) is shown in
Figure 5.3(a) and case (iii) in Figure 5.3(b).
Figure 5.3 The geometry assumed for the calculation of the
damping parameter. The X, hole pockets are indicated on the
. figures.
(a) case ( i ) of the text, the sample normal is [010], the fool] axis is parallel to the z-axis.
(b) case (iii) of the text, the sample normal is TO], the
[001] axis is parallel to the z-axis.
The Hamiltonian
To carry out the calculation we need the Hamiltonian which
describes the model system. This consists of three parts: the
spin wave Hamiltonian, the electron Hamiltonian and the
interaction Hamiltonian. Following Sparks161 the spin wave
Hamiltonian may be written:
where b' and b are spin wave creation and annihilation 9 4
operators, (Bose operators), H is the external field (including
the static demagnetizing field), A is the exchange constant, y
is the gyromagnetic ratio, q the spin wave wave-vector, and 8 9
and @ are the polar and azimuthal angles of the spin wave 9
wave-vector, with respect to the direction of as. The wave vector q should be written as a vector but will not be so
written for typographical ease. Equation ( 5 . 4 ) may be obtained
by writing the energy of the spin system including the exchange
interaction, the dipole-dipole interaction and the interaction
with the external field, and carrying out the first two
where V is the volume of the system. Sparks relates m+ to b 9
and m- to bt If this convention is used the commutator (5.5) q'
must have the opposite sign. This may be seen by comparing the
commutation relations for the magnetic moment components with
the relations for the angular momentum, t, and recalling that
8 = -YE, see ~urov[45].
The electron Hamiltonian is simply:
t Xel- = ZE c c k k k
where ci and ck are electron creation and annihilation
operators, (Fermi operators), k the electron momentum and Ek the
energy of an electron with momentum k. We need to consider
electron states located about the three cube axes [100], [OiO]
and [001]. To keep track of which states are under
t consideration we define three sets of operators: elk , clk for t the states along [1001, c2k , cpk for the states along [010],
t and cjk , cjk for the states along [0011. In terms of these
operators the electron Hamiltonian is:
The electron operators anti-commute:
The energies Elk, EZk and E3k are those of equation (5.3).
evaluated in equilibrium (mx = 0). They include the kinetic
energy and a spin-orbit shift.
The interaction Hamiltonian describes the changes in energy
which arise as the magnetization deviates from equilibrium.
Consider case (i), Figure 5.3(a). For the electron states
about the 11001 axis cos(BM) * mx/Ms and the change in energy,
A E l o o . is -(E/2Ms)mx = -(1/2)([/2MS)(m+ + m-1. For the states
about the [010] axis cos(eM) = my/Ms and A E , , , = -(E/2Ms)my
= -(1/2i)(E/2MS)(m+ - m-1. For the states about the [0011 axis
cos(eM) does not change to first order in mx and m Let Y
AE+ = -(E/2MS)m+ and AE. = -((/2MS)m-. Transforming to second
quantized notation:
where = (~Nv)exp(ik,r) is an electron wave function and
P = -(1/2)((/2MS)l/~(this symbol. P. is called 'thorn').
Similarly:
AE. = ~BZLC' c b t' k-q k q
Placing the axis labels on the electron operators:
In general,
axis. M100.
The interaction Hamiltonian is the sum AE,,, + AE,,, + AEoo,.
the component of the magnetization along the [100]
may be written:
In equi ibrium m+ and m- are zero so that MI,, = yMZ. The change
in energy for the states on the pocket at [100] for a deviation
of the magnetization from equilibrium is:
since the change in MZ is second order in m- and m+. If we let
( a - i / 3 ) = A+ and (a+i@) = A _ , with similar definitions of B+, B-
and C+ and C- for the pockets at [ 0 1 0 ] and 10011 respectively,
the interaction ~amiltonian may be written:
* * * with A+ = A_, B+ = B- and C+ = C-. This Hamiltonian is in fact
Hermitian. The constants A+, B+ and C+ are listed in Table 5-2
for the five cases of interest.
The total Hamiltmian is X = HSw + Belt+ Hint.
5.3 Calculation of the Damping
Green's Functions
Green's functions and their applications to physical
problems have been discussed in detail by ~ubarev[80]. The
reader is referred to that paper for elaboration of the
statements made in this section. For our purposes the Green's
function for two (time dependent) operators A and B is:
where the square brackets represent a commutator, p o is the
density matrix for the system under consideration when in
thermal equilibrium, the trace represents a thermal average and
8(t) is the step function, 8(t1 = 1 if t > 0, 8(t) = 0 if t < 0.
It can be demonstrated that the Green's function is the response
of the operator A to a perturbation ~6(t). The response to a
perturbation of the form Bexp(-iwt) is described by the
susceptibility ~(w):
where G(o) is the Fourier Transform of G(t.1:
since G is zero for t 5 0.
To evaluate the Green's function we differentiate equation
(5.13) with respect to time. The time derivative of the
operator A is the commutator of A with the Hamiltonian of the
system:
iKd~/dt = [A,H] ( 5 . 1 6 )
The result of this differentiation will be new Green's
functions. These may be- differentiated in turn until the
original Green's function is obtained, in which case the system
of equations resulting from differentiation may be solved
exactly, or until Green's functions are obtained which may be
related to the original function by an approximation.
The Calculation
In the calculation which follows we wish to determine the
response of the magnetization to a transverse driving field:
The perturbation is then:
Writing mx in terms of spin wave creation and annihilation
operators (5.7):
These operators enter the Green's functions as the operator
B. The operator A is that representing the component of the
magnetization of interest, mx or m or m+ or m-. For example: Y'
m- = -(1/2) (r/-v) (r/-) Z<<bm;b '9 +b 9 7' >>exp(imr)hx (5.18)
Because there is no coupling between spin waves in this
model, only the following Green's functions will be non-zero:
Then :
t m-/hx = --$iMS<<b *b +b >> = -7RMs(GI + G2) 9' -4 9
m+/hx = -yFiMs(G3 + GI)
mx/hx = -7fi~S/2 (GI + G2 + G3 + GI
my/h, = iyKMs/2(Gl + G2 - G3 - GI)
The steps in the calculation will be indicated for the Green's
function G I , the procedure being the same for the three other
functions.
Taking the time derivative of G, and multiplying ,by iK:
The commutator of b with H is: 4
so that:
where F,,(k), Fl,(k) and F13(k) are new Green's functions which
contain both electron and spin wave operators:
F 1 2 and F1, are defined in a similar manner. The first
subscript indicates which of the original Green's functions the
new Green's function is derived from. The second subscript
represents the pocket with which the electron operators are
associated.
Taking the time derivative of F1,(k) and multiplying by ifi
(and not writing down the intervening steps):
- A + < < c I b' ; b h I k-q-m '1 k m q
. Again we have new Green's functions. These may be related to
the original Green's functions by an approximation (the random
phase approximation). For example:
t where < ''1 k-q Clk-m > is the expectation value of the operator t 'lk-q 'lk-m ' t This will be zero unless m = q and < c , ~ - ~ clkUq >
= n 1 k-q' the occupation number of the state (k-q) in thermal
equilibrium. The assumption is made that the electron spin wave
interaction does not disturb the electron distribution. The
Green's functions << c , ~ - ~ t c b ;b t >> reduce to the single fk-m m q Green's function nlk-q <<b *b >> = n
9' 9 1 k-q GI. We have recovered
the original Green's functions. Performing the same contraction
on the three other sets of Green's functions in equation (5.15)
leaves:
Changing to the Fourier components of these Green's functions
(equation (5.15)) we can write the equations for the Green's
functions in a form which does not contain time derivatives,
(ifidF/dt becomes KwF). We find:
Fll(k) = p[(nlkmq - nlk)/(Hw -(Elk - k-q ) ) I (A+G, + A-G,)
(5.28)
using the same notation for the Green's functions and their
Fourier components. Substituting this expression and the
equivalent expressions for F12(k) and F13(k) into the Fourier
transformed equation for G, yields:
r 2 AND r3 are defined similarly for the pockets at [ 0 1 0 1 and
[001 I . Performing the same operations (equations 5.21 through
5.29) with G, yields:
where the fact that.A- = A and B = B has been used. q ? 9' -9 9 .
Define:
Solving for GI
GI = a
and G,:
Performing the calculation for G2 and G,:
The denominator, D, is the same for all four Green's functions.
For our case 19 = r/2, 9 %I
= r/2, so, from equation (5.4):
In the absence of spin-orbit coupling P,, P2, /3, are zero. The
denominator D and the ratios mx/hx and m /h become: Y x
my/hx = -(yfi)'MSi(w/7)/D
These expressions are the same as equations (2.18) (the
denominator of (2.18) is rewritten below) in the absence of MCA
and damping (note the difference in the coordinate system of
equation (2.18)). If 0, = 6, the denominator may be factored
as:
Such factoring will be possible if A: = A', B: = B' and C: = ~ f ,
which is true if the sample plane is a (100) plane, see .
Table 5 - 2 . If the sample plane is a (110) plane the
coefficients A+, A_, B+ and B- are related by A+ = B- and
A- = B+. In this situation p2 will equal 0, if r , = r2, that is
if the sums for the pockets at [100] and [010] are the same.
Due to the symmetry of the situation this will be true (see
Figure 5.3(a)) so that for all cases of interest 0, = P 3 c The
denominator of equation ( 2 . 1 8 ) is:
where a and y are effective magnetocrystalline anisotropy (MCA)
fields (this y should not be confused with the gyromagnetic
ratio) and G is the Gilbert damping parameter (not to be
confused with the Green's functions GI. Comparing the two
. expressions ( 2 . 1 8 ) and (5.36) suggests identifying the real part
of (PI 2 p2)/yK with an effective MCA field (possibly wavenumber
dependent) and the imaginary part with i(w/y)(G/yM,):
An interesting result of this calculation is the possibility of
the damping depending on the direction of the excursion of MS
from equilibrium, G I being the damping parameter for in-plane
excursions and G 2 being the damping parameter for out of plane
excursions. Writing the expressions for these damping
parameters out in full:
Evaluation of the Damping Parameters
To obtain values for these damping parameters we have to
evaluate a sum of the form:
for each of the three pockets. The sum is converted to an
integral :
S = .fd3k/C2a)3 [(n(k-q) - n(k))/(Kw-(E(k)-~(k-q)))l (5.40)
where n(k) is the Fermi distribution function and E(k) is the
energy of an electron in state k. Since k is of the order of
kF (=lo8 cm-'1 and q is of the order of 1/6 (=lo5 cm-'1, we may
expand n(k-q) and E(k-q) about k:
where (an/aE) = -6(E(k) - EF) assuming the Fermi distribution is a step function, and aE/ak = Kv(k). Expanding E(k-q):
The integral (5.40) becomes:
Following Heinrich, Fraitova and ~ambersky1751 the finite
lifetime of the excited electron states is taken into account by
adding a small imaginary part, K T , to the energy, where r is
the average time between collisions of an electron with a phonon
or an impurity. This term must be included on the top of the
integrand (in the An term) as well as the bottom (in the AE
term) so that:
or, writing this in terms of real and imaginary parts:
For a spherical Fermi surface this integral may be
evaluated analytically, the imaginary part of the integral
being:
where vF is the Fermi velocity and 1 is the electron mean free
path, I = v T. If ql and wr are both small compared to 1 F arctan(q1fwr) * (ql+wr) and the sum varies directly with the
relaxation time T:
If wr << 1 and ql * 1 , arctan(q1 + wr) = arctan(q1) and:
When multiplied by the appropriate constants (see equation 5.39)
this is the result of Korenman and Prange[3,4]. It is the form
of the wavenumber dependent part of the damping used in
Chapter 4 for comparison with experiment.
For a non-spherical Fermi surface, such as our pockets, the
integral must be evaluated numerically. The integral may be
evaluated in a coordinate system in which the z-axis is parallel
to the r-X axis of the pocket under consideration. This
coordinate system is different for each pocket. The direction
of q must be considered for each pocket and each case, see
Table 5-2. The energy E(k) in such a coordinate system is
(equation 5.3):
where Ex is the energy of the X, level at the X-point in the
absence of spin-orbit coupling and OM is the angle between the
equilibrium direction of the magnetization and the F-X axis.
The Fermi surface is given by E(k) = EF or:
The Fermi velocity is given by fivF = Ivk~(k) 1 :
The volume element d3k may be written:
where dSk is an element of area on a surface of constant energy
and dk is an element of length perpendicular to that surface.
Carrying out the integration over the energy the integral
becomes :
The dot product q*vF is different for each of the pockets
because we use a coordinate system in which kZ is along the F-X
axis of the pocket of interest.
5.4 Results
The summations of equation (5.39) have been carried out for
the three orientations of the magnetization in the (110) plane.
These correspond to the configurations investigated in this
thesis. Values of T of 10-14, 10-13, and 10-l2 sec,
corresponding to room temperature, 77 K, and a resistivity ratio
of 100 were used. The parameters for Nickel listed in Table 4-1
and the parameters of HSG for the hole pockets, Table 5-1, were
used. A useful conversion factor is 1 0e2 = 1.5687~10~~ ev/cm3.
The sums were evaluated for q varying from 0 to 106 cm-I which
includes the q-vectors of interest at the three temperatures
(see the numbers quoted in Table 2 - 3 ) .
The absolute value of the effective MCA field which arises
from the real part of the integrals is shown on Figure 5.4 for
the magnetization along [001] for the three values of r . The
two MCA fields, a and y, are the same for this orientation
a = y = (~2/8~S)R1(Sloo+Solo) where SloO and S o l o represent the
integrals for the pockets at [100] and [010] respectively.
Since these two pockets are equivalent:
The calculated fields for T = lo-'& see and 7 = 10-l3 sec were
150 - 12 ,740 sec
Fiqure 5.4 The effective MCA field due to the real part of the
pocket integrals, as a function of wavenumber q, for three
values of the electron relaxation time 7 . For the calculation
the magnetization was assumed to be parallel to the [0011 axis.
100 -
~=10-'~sec
-\ T=IO-'~S~C
EFFECTIVE FIELD (0e)
SO -
0 I I
0 2 4 x l o s A 1
q(cm-'>
found to be independent of q with magnitudes 137 and 128 Oe
respectively. This difference is due to the change in the
saturation magnetization, since the values of Ms were chosen to
correspond to 300 K and 99 K respectively, not to the variation
with T of the integral. There is a small q-dependence at
r = 10-l2 sec near q = 0 where qvr = w r . The fields are
negative which leads to a shift of FMR to higher field values.
The fields for the magnetization along [110] and [111] have
similar q-dependences and magnitudes. The fields are negative
for all three orientations of the magnetization.
The expressions for the two damping parameters, G 1 and G2,
for the three orientat ions of the magnetization in the [ 170 1
plane are listed in Table 5-3.
TABLE 5-3
G 1 G2
Recall that G l is the damping parameter for in-plane excursions
of the magnetization from equilibrium while G2 is the damping
parameter for out-of-plane excursions. Note that the damping
parameter G2 is the same for the three orientations of the
magnetization and that with the magnetization along [001]
- G I = G,. If the three pockets are equivalent (SloO = Solo -
So,,) then G 1 = G, and the damping for the three orientations is
the same. Values of G 1 and G, for the three temperatures at
q = 0 are listed in Table 5-4 with the values of the wavenumber
independent Gilbert damping parameter required by experiment,
see Chapter 4. Plots of GI and G2 versus q for the three
temperatures and three orientations are shown in Figures 5-5
(M, along [001]), and 5.7(a) (M, along [111]) and 5.7(b) (M s
along [110]). Also plotted on Figure 5.5 is G(0) arctan(ql)/qI
for 7 = 10-12, the form of the wavenumber dependent part.of the
damping assumed in Chapter 4. The mean free path I = 4 x 1 0 - ~ cm
was chosen to match the calculated variation of the damping
parameter as closely as possible. It may be compared with the *
value I =.5.0~10-~ cm determined using vF = Fikxw/ml. listed in
Table 5-1 . - TABLE 5-4
(001 1 ( 1 1 4 ) (110) Expt
7 G 1 G4 G 1 G2 G 1 G2 G
(sec) (lo8 sec-l)
lo-''' 0.0055 0.0055 0.0052 0.0052 0.0051 0.0055 2.45
1 0.055 0.055 0.052 0.052 0.051 0.055 7.8
0.53 0.53 0.51 0.51 0.50 0.53 14
calc
It appears that Kambersky is the only one to have attempted
ulation of the damping parameter from the known band
structure of Nickel. The numbers quoted in 1723 supersede the
earlier estimates in [23 and [ 7 1 3 . Kamberskyss calculations
were carried out for both Nickel and Iron ignoring the shifts in
Figure 5.5 The variation of the damping parameter with
wavenumber q for three values of the elecbron relaxation time 7.
- The magnetization points along [001], G I = G, for this case.
The crosses are a plot of G(O)arctan(ql )/ql where 1 = 4x10'~ cm
is an electron mean free path at a = sec.
. Fiqure 5.6 The variation of the damping parameters GI and G2
with wavenumber q for three values of the electron relaxation
time r . (a) The magnetization points along [ 1 1 1 ] .
(b) The magnetization points along [ 1 1 0 ] .
energy levels due to spin-orbit coupling. Calculations were
also made for the states around r - X including the energy shifts.
These results are quoted in his Table IV which is reproduced
here:
G is the Gilbert damping parameter for Nickel when Ms points
along a ( 1 1 1 ) direction, and for q = 0. These numbers may be
'compared with those in .Table 5-4.
The damping assumed-in Chapter 4 for comparison with
experiment was
where oo was the dc conductivity, p the resistivity and l D the
d-electron mean free path. The values of a and b required to
fit the FMAR data of Cochran and ~einrich[37] were
a = 1.07x108 sec-' and b = 1.19x108 sec-l. With a resistivity
ratio of 100 (r=10-'~) the spin-flip damping, b, is negligible
and the damping would be:
When this expression is evaluated for q approaching zero the
resulting value for G is some 200 times the values listed in
Table 5-4.
The predicted anisotropy of the linewidth is not in
agreement with experiment. For example, using G,(~=O) and
G2(q=O) for our comparison, it would be expected that AHloo
should be larger than AH,,, by approximately 4%. Since G 1 is
not equal to G2 for Ms along ( 1 1 1 ) the predicted anisotropy
would have to be determined by carrying out a calculation of the
absorption which included the two damping parameters GI and G 2 .
This has not been done, however it is unlikely that the
anisotropy in the linewidth would exceed 4%. In any event the
anisotropy is opposite to that observed experimentally, since
AH1 was found to be greater than AHloo by approximately 12%.
The damping due to the X, hole pockets calculated using
this simple model of interacting electrons and spin waves is
unable to account for the magnitude or the anisotropy of the
damping observed in experiment. We conclude that other portions
of the Fermi surface must play a more important role in the
magnetic damping in Nickel at low temperatures than has been
hitherto recognized.
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